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: .


. . - . . [102, 103], . . [46, 47], . . [35, 36], . ­ . , , L -, . . § 1 , . , , , , , . . . . [21, 22, 54, 107]. § 2 [75] . . , O N lo g 2 N

(

)

, N ­

, . § 3 ­ . . § 4 , ()

Ax = f ,

(0.1)

3


: .

A ­ N . A , . .

( A + B) x = f

,

( 0 .2 )

, B ­ N . , B , , (0.2): 1) f
N



> 0 x = x ( f ) ; 2) f R ( A)
x 0 (0.1). A , (0.2) (0.1). B , A . A , B , 4.1 ­ 4.3. , (0.2)

( A, B, , f )



(

A A, E , , A f , A ­ - -

)

A , E ­ , > 0 , . . [104, 105];

(

A A, L L, , A f

)

,

L - [60, 61, 63];

( A, E , , f )

,



A = A 0 ,

> 0,





.. [48, 49, 50], A . . [108];

( A, E , i , f )

, A = A 0 ,

> 0 , i - , ..

4


: .

[7 ­ 11]; B =

m =1



k

g m em , em , g m , m = 1, k ­

k er A k e r A , , . . . . [98 ­ 100, 106] , . . , . . . . [3 ­ 5, 13, 14, 52 ­ 55]. § 5 , B = E ­

( A + E ) x = f

.

( 0 .3 )

(0.3) : 1) . . [108], A , f ; 2) . . [48] . . [8], ( , ) A . , , (0.3) (0.1) , A . , , , (0.3) (0.1), A . § 6 (0.2) (0.1) 0 . , 0 B . § 7

( A + B) x = f

,

( 0 .4 )

, , .

5


: .

§ 8 , [25]. ( « ; ). [15, . 163]. § 9 , A N N C , A + C . § 10 , (0.1). . -, R ( A ) R A , x

()


-

­ , ­ ;

(f)

(0.2)

0 R0 R ( A ) , , ; -, A B , (0.2) (0.1), A B , (0.2) (0.1) .. 13 , (0.2) (0.1). B , (0.1). § 11 ­ , . , .

6


: .

§ 12 , . , L -. § 1

(S ; f , g )



(V ; f , g ) L

-.

) ( S ; f , g ) , ; ) (V ; f , g

)

.

(V ; f , g ) ( S ; f , g ) 0 . , A L k e r A k e r L = {0} , , ­ . (V ; f , g ) , . § 2 L -

(

S ; f , g ) (V ; f , g ) . -

, . §§ 3 ­ 4 L - . . , :

A- A µ ,

L-L µ ,

- f f , f

g -g g ,

µ > 0 , > 0 ­ , . -

f ­ x A, L, , g

(

)



7


: .

­ x

0

( A, L, f , g )

, ­ -

. . . . [2], . . [56, 57], . . [110 ­ 112] . , , . § 1 , . §§ 2 ­ 4 , . §§ 5 ­ 6 . §§ 7 ­ 8 . § 9 I II , . II I , . §§ 10 ­ 11 I II . 4 , . , , , , . - ,

8


: .

- , , . .. . 10­01­00297­.

9



:

1.

§1

1
1. 1.1. . N , , , . , , . N

a0 a- a - ... a- a -

a1
1 2

a a

2

...a ...a ...a

N -2 N -3 N -4

a a a

N -1 N -2 N -3

a a

0 -1

a1
0

N +2 N +1

... a- a

N +3

... a- a

N +4

...... ... a0 ...a
-1

... a1 a
0

- N +2

- N +3

.

. [21, 23, 52]. 1 .1 . ,

A = a

mn



N âN

, -



a

m n

=a

m n

(1.1)

1 m, n, m, n N ,

m - n = m - n .


(1.2)

, ,

( m, n)
m n



( m, n)

(1.2),

a

m n

a

, .
9


:

1.

§1

1.2. . A = a



mn



N âN

, (1.1)

m - n m - n ( mo d N ) .
N

(1.3)

a0 a N a N ... a2 a 1

a1
-1 -2

a a

2

...a ...a ...a

N -2 N -3 N -4

a a a

N -1 N -2 N -3

a a

0 N -1

a1
0

... a3 a
2

......... a 4 ... a0 a
3

... a1 a
0

...a

N -1

.

, , , . 1.3. . N

Q=


0 1 ... 0 . 0 0 ... 1 1 0 ... 0


(1.4)

E

n





n : 1 n N ,
0 Q= E1
10

E

N -1

. 0

(1.5)


:

1.

§1



0 Qk = Ek


E

N -k

, 0

(1.6)

QN = E ,
E = E
N

(1.7)

­ N . ,

N = 4
Q= 0 1 0 0 0 0 1 0 , 0 0 0 1 1 0 0 0 Q2 = 0 0 1 0 0 0 0 1 , 1 0 0 0 0 1 0 0

Q3 =

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

,

Q4 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

=E.

(1.6) (1.7) . Q
N -1

0 = EN

-1

E1 T = Q , «» ­ 0

,

Q TQ = E ,

Q T = Q -1 .

(1.8)

1.4. . .
1 2 1 1.1. C ( ) + C ( ) C ( )

C ( 2) N N .
11


:

1.

§1

1 2 (1) (2 . C ( ) = cmn C ( ) = c mn) ­ -









. c

(1) m'n'

=c

(1) m '' n ''

c

( 2) mn

=c

( 2) m '' n ''

,

m '- n ' m "- n "( mo d N ) ,
c
mn

1 m ', n ', m ", n " N (1.9)

(1) ( = c mn + cm2n) . (1.9)

c

m'n '

(1 ( (1) ( = c m ')n ' + c m2')n ' = cm " n " + cm2")n " = c

m"n"

.

. 1.2. C C N N . . . 1.2. , c0 c1 . . . c
N -1



C ,

C = c0 E + c1Q + . . . + c

N -1

Q

N -1

=

N -1 k =0



ck Q k .

(1.10)

. . C ­ c0 c1 . . . c
N -1

.
...c ...c ...c
N -2 N -3 N -4

c0 c N c C= N ... c2 c 1

c1
-1 -2

c c

2

c c

0 N -1

c1
0

... c3 c
2

......... c 4 ... c0 c
3

...c

N -1

c N -2 c N -3 = ... c1 c0 c
N -1

12


:

1.

§1

c0 0 0 = ... 0 0

0 c0 0 ... 0 0 0 c0 ... 0 0 ... ... ... ... ... 0 0 ... c0 0 0 0 ... 0 c0 0 0 ...0

0 0 0 + ... 0 c 1

0 0 c1 ... 0 0 0 0 ... 0 0 + ... ... ... ... ... 0 0 ... 0 c1 0 0 ... 0 0 c1 0 ...0

0 c N 0 ... + ... 0 0

0
-1

0 0
N -1

...0 ...0 ...0

0 c ... 0 0

0

......... 0 ...0 0 ...c
N -1

0 0 = ... 0 0 c
N -1

= c0 E + c1Q + c2Q 2 + . . . + c

N -2

Q

N -2

+c

N

Q -1

N -1

=

N -1 k =0



ck Q k .

. .

Q 0 = E , Q, Q 2 , . . . , Q

N -1

, 1.1 1.2. . . 1.5. . . 1.3. A N , Q ,

A Q = QA .
13

(1.11)


:

1.

§1

. . A , (1.10). k Q k Q = Q Q k ,

AQ =

N -1 k =0



ck Q k Q =
k

N -1 k =0

ck (

Q kQ =

)

=

N -1 k =0

ck (

QQ

)

=Q

N -1 k =0



ck Q k = QA .

.

­ , (1.11). , A mn

. A = a

. (1.11)

A = QAQ T .

(1.12)

A , :

a11 A= A21
, (1.5) (1.8),

A1 2 A22

A2 2 QAQ T = A12

A2 1 , a1 1
2

(1.13)

A1 2 = a1

2

a2 a 21 ... a1N , A21 = , A2 2 = a a N1 N

2

. ... a NN ...a
2N

14


:

1.

§1

m - n - (1.12) a :
mn

, -

a

m +1, n +1



m < N, n < N ;

a

m +1,1

m < N, n = N ; m = N, n < N ; m = N, n = N .

a1, a1,

n +1

1

(1.12)

a
[ mo d N

m, n

=a

m +1, n +1[ mo d N

]

,

(1.14)

]

­ -

N . (1.14) (1.3), A . . 1.6. . , . 1.1. .
1 2 . C ( ) C ( ) 1 1 . 1.3 C ( )Q = Q C ( ) ,

C ( 2)Q = Q C ( 2) . ,

(

C (1)C ( 2 ) Q = C (1) C ( 2 )Q = C (1) QC ( 2 ) =

)

(

)

(

)

= C (1)Q C ( 2) = QC (1) C ( 2) = Q C (1)C ( 2) .
15

(

)

(

)

(

)


:

1.

§1

1 2 1.3 , C ( )C ( ) .

1.2. , . . C . C Q = Q C C
-1
-1

,

C

( C Q ) C -1 = C -1 ( Q C ) C

-1



QC

-1

= C -1Q .
-1

1.3 C

.

1 2 1.3. C ( ) C ( ) ­ .

C (1)C ( 2 ) = C ( 2 )C (1) .
. 1.2
N -1 N -1 k =0

(1.15)

C (1) =



k =0

( c k1)Q k ,

C ( 2) =



( c k 2 )Q k ,

(1 (2 ck ) , ck ) , k = 1, N - 1 ­ , 1 2 C ( ) C ( ) , .

CC

(1) ( 2 )

=

N -1



k =0



1 m, n N m + n k ( mo d N



cm c
)

(1) ( 2 )
n

k Q ,

CC

( 2 ) (1)

=

N -1



k =0

1 m, n N m + n k ( mo d N



c m cn Q k . )
( 2 ) (1)

16


:

1.

§1



1 m, n N m + n k ( mod N



(1 ( c m ) cn2 ) = )

1 m, n N m + n k ( mod N



(( cm2)c n1) )

(1.15). 1.7. N - . N - [45, . 127 ­ 129]:
2 m N

wm = e

-i

,

m = 0, N - 1 .

(1.16)

1.4. m : m = 0, N - 1
N wm = 1,

m = 0, N - 1 .

(1.17)

,
N wm

= e

-i

2 m N

=e

N

(

- i 2 m

)

= 1m = 1,

m = 0, N - 1 .

1.5. k , m : k , m = 0, N - 1
k m wm = wk = w1km .

,
k wm

= e = e

-i

k 2 m N

=e
m

-i

2 km N

,

m wk

-i

2 k N

=e
km

-i

2 km N

,

w1km

= e

-i

2 N

=e

-i

2 km N

.

17


:

1.

§1

1.6. m : m = 0, N - 1
N -1 k =0



N , k wm = 0,



m = 0, m 0.

(1.18)

, m = 0 , w0 = 1 , ,
N -1 k =0



k w0 =

N -1 k =0


N

1k = N ;

1 m N - 1 , (1.17) wm = 1, wm 1 , ,
N -1



k =0

k wm

=

N 1 - wm

1 - wm

=

1-1 =0. 1 - wm

(1.18) . 1.8. . . N -

1 F= N

0 w0 0 w1

w1 0
1 w1

2 w0

w12
...

...

...

w

0 N -1

w1 N

-1

w

2 N -1

N ... w0 -1 ... w1N -1 , ... ... N -1 ... wN -1

(1.19)

wm , m = 0, N - 1 (1.16), ( ). 1.1. , (1.19) , [21, 23]. 1 / N . .

18


:

1.

§1

1.4. (1.19) :

F = F 1 F = FT = N
0 w0 0 w1 1 w1

-1

,
0 N -1

(1.20)



w1 0 ...
N w0 -1

... w1N
-1

... w1 -1 N ... ... N -1 ... wN -1 ...w

­ - F . .

S = s

mn

= F F .

(1.21)

1.4 ­ 1.6,

s

mn

=

1 N
N -1 k =0

N -1 k =0



mn wk wk =

1 N 1 N

N -1 k =0 N -1 k =0



w1k m w1k n =
n-m

= =

1 N



w1-

km

w1k n =



w1k (

)

=

1 N , m = n, 1, = N 0, m n, 0,

m = n, m n.

, , (1.21) : S = E . (1.20). . 1.9. . (1.4) N - (1.16). ,

d e t ( Q - E ) = ( - 1)
19

N

(

N - 1) ,

(1.22)


:

1.

§1

N ­ Q . ,

m = wm ,

m = 0, N - 1 .

(1.23)

, , :

em =

0 wm w1 m , N wm - 2 N wm -1

m = 0, N - 1 .

(1.24)

,

m

e

m

-

(1.23) (1.24). m = 0, N - 1 ,

Q em =

0 1 0 ... 0 0 0 1 ... 0 0 0 0 ... 1 1 0 0 ... 0



0 wm w1 m = N wm - 2 N wm -1

=

w1 w1 m m 2 2 wm wm = N N wm -1 wm -1 N 0 wm wm

0 wm w1 m = wm N -2 wm N -1 wm

= m em .

, .

20


:

1.

§1

1.5. (1.23) (1.24), . N - . , ( ) N . , . , [1, 12] . , , . , :
m

em , -

d i a g ( a1 , a 2 , . . . , a

N

)

=

a1 0 0

0 a
2

0

0 ... 0 . ... aN ...

(1.25)

1.6. Q F ­ N , .

F Q F = d i a g ( w0 , w1 , . . . , w

N -1

)

,

(1.26)

wm , m = 0, N - 1 N - (1.16). .

1 F Q= N

w w w

0 N -1

0 w0

0 w1


m N -1


m w0

w1m w1N -1


N -1 N -1


N w0 -1

m ... wN - 2 = N- ... wN -1 2 ...w
0 N -2

21


:

1.

§1

1 1 1 =diag , , .. . , F = w w wN -1 0 1

= d i a g ( w0 , w1 , . . . , w


N -1

)

F ,

F Q F = d i a g ( w0 , w1 , . . . , w = d i ag ( w0 , w1 , . . . , w
.
N -1

N -1

)

F F =
N -1

)

E = d i a g ( w0 , w1 , . . . , w

)

.

1.1. k : k = 0, . . ., N - 1
k F Q k F = d i a g w0 , w1k , . . . , w

(

k N -1

)

.

1.10. . , , . , ­ . 1.7. c = c0 , c1 , . . . , c
T

(

N -1

)

­

C . : 1) C

m = ( wm ) ,


m = 0, N - 1 ,
N -1 k =0

(1.27)

( w) =
2)



ck w k ;

(1.28)

F C F = d i ag ( 0 , 1 , . . . ,
22

N -1

)

,

(1.29)


:

1.

§1

Fc =

1 ( , , .. ., N01

N -1

)

T

.

(1.30)

. 1.2

C=
1.1

N -1 k =0



ck Q k .

F C F = F = = diag
N -1 k =0

N -1



k =0

ck Q k F =

N -1 k =0



ck F Q k F =
k N -1



k c k d i a g w0 , w1k , . . . , w N -1 k =0

(

)

= =

N -1 k =0



k c k w0 ,



ck w1k , . . . ,

N -1 k =0



ck w

k N -1

= d i ag ( w0 ) , ( w1 ) , . . . , ( w

(

N -1

))

,

( w ) (1.28). (1.27) (1.29) . (1.30) . . 1.11. . ()

Cx = f ,
C ­ N ,

(1.31)

f

N

­ ,

­ . (1.31)

x

N

F C F y = F f


(1.32) (1.33)

x = Fy .
23


:

1.

§1

(1.32)

F C F . . 1.7
F C F .



( w ) = c0 + c1w + c2 w 2 + . . . + c
c0 , c1 , c 2 , . . . , c C .
N -1

N -1

w

N -1

,

(1.34)

­ -

C

­ , wm , m

()

0, N 1

w

, m = 0, N - 1 (1.32)


F f m ym = , m = 0, N - 1 , ( wm )
y
m

(1.35)

F f

m

m - y F f ,

. x
m

x ­ (1.33)



1 ( w ) =

1 y 0 + y1w + y 2 w 2 + . . . + y N

(

N -1

w

N -1

)

(1.36)

w = wm , m = 0, N - 1 . , ( w ) , 1 ( w ) w = wm , m = 0, N - 1 (1.35). (1.35) N , (1.34) (1.36)

w = wm , m = 0, N - 1 -

24


:

1.

§2

N 2 ( ) . () N l o g 2 N .

2. 2.1. . N [25, . 301 ­ 302], , , . , , . N

b0 b1 b N b N

b1
-2 -1

...b

N -2

b b

N -1 N

... b2
N -2

b
N

b2 b2

N -2 N -1

.

. 2 .1 . ,

B = bm n

NâN

, -



bm

n

= bm

n

(2.1)

1 m, n, m, n N ,

m + n = m + n .


(2.2)

, ,

( m, n)



( m, n)

(2.2),

25


:

1.

§2

a

m n

a

m n

, -

. 2 .2 . .

B = bm n

NâN

, (2.1)



m + n m + n ( mo d N ) .
. « » . [75] « ». N

p0 p1 pN pN

p1
-2 -1

...p

N -2

p p p

N -1 0

...p
N -3

p
0

N -3 N -2

p

.

, , , . 2.3. . N

0 0 R = 0 1

... 0 1 0 . 0 0 ... 0 0 0

(2.3)

26


:

1.

§2

() , N . . N :

R2 = E , RT = R , R -1 = R .

(2.4) (2.5) (2.6)

(2.4)-(2.6) . R B = bm n





NâN

,

B

0 ... 1 b1 1 ... b1N R B = 1 ... 0 bN 1 ... bNN

bN 1 ... bN N = b 11 ... b1N

,

(2.7)

, B

b1 1 ... b1N B R = b N 1 ... b NN



0 ... 1 b1N = 1 ... 0 bNN

... b1 1 . ... bN 1

(2.8)

2.4. . . , (2.7) (2.8) . 2.2. P N , R P P R .

27


:

1.

§2

, R P P R . 2.2 : 2.3. C N , R C

C R .
2.2 2.3 , . 2.5. . .
1 2 1 2.1. P ( ) + P ( ) P ( ) 2 P ( ) N N . 1 . 2.2 R P ( )

R P ( 2 ) .

R P (1) + R P ( 2) = R P (1) + P ( 2 )

(

)

. 2.3 ,
1 2 P ( ) + P ( ) . . 1 2 2.2. P ( ) - P ( )

P (1) P ( 2) N N .
2.3. P P N N . 2.1. 2.4. .

28


:

1.

§2

1 2 . P ( ) P ( ) ­ . 1 2 2.2 P ( ) R R P ( ) . 1 2 ( 1.1), P ( ) R R P ( ) ­ . (2.4) ,

P (1) R R P ( 2 ) = P (1) R 2 P ( 2 ) = P (1) E P ( 2 ) = P (1) P ( 2 )
. . 2.5. P ­ C ­ . P C C P . . 2.2 R P . R P C . 2.3

R ( RP C ) = R 2 PC = E PC = PC
. P R ­ , C P R .

(C P R ) R = CP R 2 = CPE = CP
. . 2.6. , . . P ­ . R P . 1.2 ( R P

)

-1

. ,

( R P ) -1 = P -1R -1 = P -1R
2.2 P .
-1

. -

29


:

1.

§2

2.6. . . :

p d ( µ 0 , µ1 , . . . , µ

N -1

)

=

µ

0

0 0

0 0

µ

N -1

0 µ1 . 0 ...

(2.9)

2.4. R ­ (2.3) F ­ (1.18) N .

F R F = p d ( w0 , w1 , . . . , w

-i 2 k N

N -1

)

,

(2.10)

wk = e

,

k = 0, N - 1 .

(2.11)

. F R F = s



mn

,

s 1 N
N -1 k =0

mn

=

1 N

N -1 k =0


)

m wk w

n N -1- k

= w1- k (
m+ n

=



w1-

km

w1n (

N -1- k

=

1- w N1

n

N -1 k =0



)

=

w m , m + n = 0 ( mo d N ) , = 1 = m + n 0 ( mo d N ) , 0,
w , m + n = 0 ( mo d N ) , = m m + n 0 ( mod N ) , 0,
.

30


:

1.

§2

2.7. . . . 2.5.

-

(

p 0 , p1 , . . . , p

N -1

)

­ -

P ,

F P F = p d w0 ( w0 ) , w1 ( w1 ) , . . ., w

N -1 k =0

(

N -1

(w

N -1

))

,

(2.12)

( w) =
p d (...) ­ (2.9).



p

N -1- k

wk ,

(2.13)

. (2.4)

F PF = F P R 2 F = F




( P R ) F F RF


.

(2.14)

(

PR
N -2


0

p

N -1

,p

, . . . , p1 , p
F


)





. 1.7
N -1

( PR ) F = d i ag ( ( w0 ) , ( w1 ) , .. . , ( w

))

,

(2.15)

( w ) (2.13). (2.10) (2.15) (2.14), (2.12). . 2.8. .

Px = f ,
P ­ N , f
N

(2.16) ­ ,

x

N

­ .
31


:

1.

§2

(2.16)

F PFy = F f


(2.17) (2.18)

x = Fy .


(2.17)

F P F . . 2.5
F P F p d (...) - (2.9).

( w) = p

N -1

+p

N -2

w+ p

N -3
N -3

w 2 + . . . + p1w
,p
N -2

N -2

+ p0 w

N -1

,

(2.19)

p0 , p1 , . . . , p

,p

N -1



P .

P ­ , wm wm ,

()

m = 0, N - 1 (2.17)


y0 =
y F f

F f F f 0 N -m ; ym = , m = 1, N - 1 , w0 ( w0 ) wm ( wm )

(2.20)

m

m

m - y F f ,

. x
m

x ­ (2.18)



1 ( w) =

1 y 0 + y1w + y 2 w 2 + . . . + y N

(

N -1

w

N -1

)

(2.21)

w = wm , m = 0, N - 1 ­ N - (2.11).

32


:

1.

§3

, ( w ) 1 ( w

)

w = wm , m = 0, N - 1 ,

(2.20). (2.20) N , (2.19) (2.21) w = wm , m = 0, N - 1 N 2 ( ) . N l o g 2 N .

3. 3.1. . ( 2.2), P ­ , P R . P x = f P C y = g

C . C P : C = P R , R ­ (3.3). g f : g = R f . ,
, , . , 1, . A x = f , A C P : A = C + P . P . C . C x + P x = f ,
33


:

1.

§3

. . 3.2. . , . C ­ . P ­ , A = C + P ­ . A1 A2 ­ , A1 + A2 , A1 - A2 A1 A2 . . « » . « ». C P

(

c0 , c1 , . . . , c

N -1

)



(

p0 , p1 , . . . , p

N -1

)

,

(3.1)

. :

( w) = ( w) = w

k =0



n -1

c k w k , m = ( wm ) , m = 0, N - 1 , w k , µ m = ( wm ) , m = 0, N - 1 .

(3.2)

k =0



n -1

p

N -1- k

(3.3)

3.1. C P ­ , (3.1), .

F



( C + P ) F = di ag ( 0 , 1 , .. . ,
m

n -1

)

+ p d ( µ 0 , µ1 , . . . , µ

n -1

)

,

m µ

(3.2) (3.3).

34


:

1.

§3

(1.28) (2.12). F


(C + P ) F

-

. N = 2 M + 1 ­ ,

F



(C + P ) F =
µ1 , 2 M

0 + µ =

0

1


µ


M

µ

M

M +1

M +1

µ

2M

N = 2 M ­ ,

F



(C + P ) F =
µ
.

0 + µ =

0

1


1

µ


M -1

µ M + µ
M

M -1

M +1



M +1



µ

2 M -1



2 M -1

, , .
35


:

1.

§3

3.3. .

Ax = b ,

(3.4)

A = C + P ­ N , 2.1. (3.4)

F




( C + P ) Fy = F b
x = Fy .

(3.5) (3.6)

(3.5) . N . N = 2 M + 1 ­ , (3.5) :

(

0 + µ

0

)

y 0 = F b

0

M :

y +µy = F b , k N -k k kk N - k y k + µ N - k y N - k = F b N

k = 1, M ;
-k

,

N = 2 M ­ , (3.5) :

(
(

0 + µ

0

)
)

y 0 = F b , 0 y
M

M + µ

M

= F b

M

M - 1

y +µy = F b , k N -k k kk N - k y k + µ N - k y N - k = F b N
36

k = 1, M - 1 .
-k

,


:

1.

§4

:

y = ( y 0 , y1 , . . . , y

N -1

)

T

,

F b = F b , F b , . . . , F b 0 1

(

N -1

)

T

.

, N (3.5) N l o g 2 N . x (3.6) ( ). , (3.4) N l o g 2 N .

4. 4.1. . . , .

Ax = f ,
A ­ N , f
N

(4.1) ­ -

, x N ­ , N N - . (4.1)

( A + B) x = f

,

(4.2)

B ­ N , ­ . , (4.2) (4.1), , -, (4.2) f
N

37


:

1.

§4

x , x




> 0 , -

0

(4.1), f R ( A ) . , A , A + B B 0 . x0 ­ (4.1),

x - x0 = O ( ) .

(4.3)

, A , (4.2) (4.1) B N . A , , -, (4.1) f n , -, (4.1) (4.2) (4.3).

( A, B )

A B , (4.2) , (4.2)

(4.1).

( A, B, , f )



(

A A, E , , A f , A ­ - -

)

A , E ­ , > 0 , . . [104, 105],

(

A A, L L, , A f

)

,

L - [60, 61, 63],

( A, E , , f )

,



A= A 0,



> 0,





.. [48 ­ 50], A . . [108]; ( A, E , i , f ) , A = A 0 , > 0 , i , .. [7 ­ 11]); B =

m =1



k

g m em , em , g m , m = 1, k ­

38


:

1.

§4

k er A k e r A , , . . . . [98 ­ 100, 106] , . . , . . . . [3 ­ 5, 13, 14, 52 ­ 55]. 4.2. . : A + ­ A [12, 22, 25]; Q = E - A + A ­ k er A ­ A ;

P = E - AA

+

­




0



k e r A ;
0

F = P BQ ;

T = F + B - E A+ ; 0 = {z : 0 < z <

(

)

}

,

­ -

; ( ) = d et ( A + B ) . . 4.1. (4.2) (4.1), : () 0 ,

( 0)

0

R ( A ) R ( B Q ) = {0} .
4.1 . 4.4. () : ()

(4.4)

F +F = Q ;
( )

(4.5)

FF + = P ;
( )
39

(4.6)


:

1.

§4

( A + B ) -1 = - 1F + +
C , 0 < < T B ; ()
-1

m =0


)



m (T B ) T ( B F + - E ) ,
m

( 4 .7 )

, = (T B

­ -

( A + B)
( )

-1

=O

(

-1

)

, 0 ;

(4.8)



( A + B)
()

-1

con st, 0 ;

(4.9)



( A + B ) B cons t, 0 ;
()

-1

(4.10)

( A + B ) -1 A co n s t ,
( )

0

;

4.11)

R ( A) + R ( BQ ) = n ;
()

(4.12)

(k ) (0) 0 ,
k = n - r , r = r an k A ­ A .

(4.13)

()-() (), 4.1, . 4.2. ()-() . 4.2 . 4.3.

40


:

1.

§4

4.2 , 4.1 () ()-(). ()-() . A B

A11 A= A21

A12 , A2 2

B11 B= B2 1

B1 2 , B2 2

(4.14)

A1 1 B1 1 ­ r , r = r an k A . 4.3. A , B , A1
1

­ r , r = r an k A ,

d e t A1 1 0 . (4.2)
(4.1) ,

d e t B22 - A2 1 A1-11B1 2 - B21 A1-11 A1 2 + + A2 1 A1-11B1 1 A1-11 A1
2

(

)

0.

(4.15)

4.3 . 4.4. 4.3. 4.2. , , . 4 .3 .1 . ( ) ( ) .
N



:
N

N

- 0

­

. - 0 , : 1) x 0 x N ; 2) x = x x 0 .
0

-

0 . , x = 0 x 0 .
41



:

1.

§4

4.1. 1 , 2 :

N



N

­ -

0 .

1 = 2 ,











k e r 1 = k er 2 ,
k e r k = x N : k x = 0 , k = 1, 2 .
. . . k e r 1 = k e r 2 0 . x

N

{

}

-

x = x1 + x 2 , x1 0 , x 2 0 . ,

1 x = 1 x1 = x1



2 x = 2 x1 = x1 .

, 1 x = 2 x x N , 1 = 2 . . 4.2. 1 ,
N 2

­ -



1 2 = {0} ,


di m 1 + dim 2 = N .

(4.16)

= {0} , 1 2 = N . 1 2
. (4.16)

(4.17) (4.18)

1 2 = N .

(4.19)

42


:

1.

§4



z , z ( 1 1 2

2

)



. (4.19)

(

1

2

)



= {0} . , z = 0 . (4.17) .

, (4.17)

di m = N - di m k , k
(4.19). .

k = 1, 2

() (). 4.1 () ,

k e r Q = k er F + F .
k e r F + F k e r Q , . , z k er F + F . F z = = P B Q z = 0 , Q z k e r P B = R ( A ) . (4.4) , B Q z = 0 . A Q z = 0 ,

( A + B ) Qz = 0



. ( 0 ) = d e t ( A + 0 B ) 0 ( ()) , Qz = 0 ,
z k e r Q . () () . () (). (4.4). y R ( B Q ) R ( A ) . z
N

,

y = B Q z R ( A ) = k e r P . P y = P B Qz = 0 . (4.5)
4.1 , Q z = 0 . y = B Q z = 0 , ,

R ( B Q ) R ( A ) = {0} . (4.4) .
0 , .

( 0)

0 . -

( ) = det ( A + B ) = 0 .



( A + B) x = 0




x . ( A + B ) x = 0 x

= 1 1 .
43


:

1.

§4

, k 0

k , x



k

x0 k , x0 = 1 .

(

A + k B ) x



= 0, k = 1, 2, . . . ,
k

(4.20)

k , A x0 = 0 , , x0 k e r A . P A = 0 , (4.20)

P ( A + k B ) x



k

= k P B x



=0.
k

k 0 , P B x



=0.
k

k , P B x0 = 0 . Q x0 = x0 , 0 = P B x0 = P B Q x0 = F x0 , , F F x0 = 0 . (4.5) : F F x0 = Q x0 = x0 = 0 . x0 : x
0 + +

= 1 . () ().

() () . 4.3.2. () (). () (). 4.1 (4.5) , k e r Q = k e r F . Q F ­ , ,

di m ker Q = dim k er F = di m ker F .
, d i m k e r Q = d i m k e r P ,

d i m k e r P = d i m k er QB P .
, k e r P k e r Q B P = k e r F , k e r P = k er F . 4.1 (4.6). () () .
44


:

1.

§4

() (). 4.1 (4.6) , k e r P = k er F . P F ­ , di m ker P = d im k er F . , d i m k e r P = d i m k e r Q ,

di m ker Q = di m ker PB Q .
, k e r Q k e r P B Q , k e r Q = k e r F . 4.1 (4.5). () () . () () . 4.3.3. () (). . 4.3. (). :

AF + = 0 , AT = - AA+ .

(4.23) (4.24) (4.25) (4.26)


A T B F + - E + BF + = E , PB T = 0 .
. A Q = 0 , F + = F

(

)

(

FF
,

+

)



F = ( P BQ


) = Q B P = QB P
FF
+

AF + = AF



(

)

= AQB P FF

(

+

)

=0.

(4.23) . :

A T = A F + B - E A + = AF + BA + - AA + = - A A + .
45

(

)


:

1.

§4

(4.24) . (4.23) :

A T B F + - E + BF + = - A A = E - AA

(

)

+

(

BF + - E + BF =

)

(

+

)

B F + + A A + = P BQF + + A A + = F F + - Q + E = E .

(4.25) . (4.6)

PB T = PB F + B - E A + = P BF + B A + - PB A + = = PB QF + B A + - PB A + = FF + - P BA + = 0 .
(4.26) . . () (). ,

(

)

(

)

( A + B ) -1F + +




m=0





m (T B ) T ( B F + - E ) =
m



= -1 A F + + A T B F + - E + B F
+

((

)

+

)

+

m =1






m ( AT + E ) ( B T )

m

(

BF + - E .

)

(4.27)

, 4.3,

( A + B ) -1F + +

m=0





m (T B ) T ( B F + - E ) = E .
m



(4.28)

(4.7). () () . () (). (4.7), (4.28). (4.27) (4.28) . , .

46


:

1.

§4

. , , ,

A T B F + - E + BF + = E .

(

)

(4.29)

. (4.24),

A T B F + - E + BF + = - A A

(

)

+

(

BF + - E + BF + =
+

)

= - A A + BF + + AA + + BF + = E - A A = PB QB P F F = FF


(

)

BF



(

FF

+

)

+ AA + =

(

+

)

+ A A + = P BQ 2 B P F F
+

(

+

)

+ AA+ =

(

FF

)

+ AA + = FF + + AA + = E .

P = E - A A + , (4.6). () () . () () . 4.3.4. (), (), () (). ()-() ()-() : 1 ) ( ) ( ) ; 2) () (); 3 ) ( ) ( ) ; 4) () (); 5) () ().

( A + B) ( A + B)

-1 -1

.

A ­ ,

0 . () -

, ( A + B
47

{

)

-1

}

-


:

1.

§4

. , .
mn









( A + B ) -1
a

-1

:
mn

( )

­ -

+ bmn A + B .
= N1 ( ) ( ) ... ( ) ( )
11

( A + B)

-1



( ) ( )
1N

. NN ( ) ... ( )





mn

( ) / ( )
-1

-s

mn

,

( A + B ) -1

-s

,

s = ma x s


1 m , n N m n

. A , s 1 . . :

( A + B ) -1

? ­ . 4.4. A , B ­ N k = N - r , r = r an k A .

( 0 ) = ( 0 ) = . . . = (

k -1)

(0) = 0

.

(4.30)

. ; j1 , . . . , j

(

m

)

, m = 1, k -

N , ( ) j s - a
ij
s

+ bi j bi
s

j

s

(

i = 1, n, s = 1, m .

)

, (

)

( m ) ( ) = m!

1 j1 < . . . < jm n



( ; j1 ,. . . , j

m

)

.

(4.31)

48


:

1.

§4

(4.30) r . A , r < n . r = N - 1 . k = 1 (4.30) ( 0 ) = 0 , : ( 0 ) = d e t A = 0 . r = N - 2 . k = 2 (4.30)

( 0 ) = ( 0 ) = 0 . . . (4.31)

( 0 ) =

1 j n

( 0; j )

.

(4.32)

( 0; j ) ­ N - , A j - j -

B j = 1, N . ( 0; j ) j - : ( 0; j ) =
Aij ­

(

)

( -1)
i =1

n

i+ j

bij Ai j ,

( N - 1)

- A , -

ai j , bi j ­ B

(

i, j = 1, N . r = N - 2 ,

)

( N - 1)

- A ; Aij = 0 i, j = 1, N . , , r = r0 < N - 2 . k > 2

(4.32) .

r A . ,

( m ) ( 0 ) = 0,
(4.31)

m = 1, k - 1 .

(4.33)

( m ) ( 0 ) = m!

1 j1 < . . . < j m n



( 0; j1 , . . . , j

m

)

.

49


:

1.

§4

,



( 0; j1 , . . . , j

m

)

=0





m = 1, k - 1



1 j1 < . . . < j m N . m = 0, k - 2 .

(

k -1)

( 0 ) = ( k - 1)!

1 j1 < . . . < j



k -1

N

( 0; j1 , . . . , j

k -1

)

=0.


( 0; j1 , . . . , j


k -1

)







. j1 -

, , B . ( N - 1) - , k - 2 B . j 2 , , B . ( N - 2 ) - , k - 3

B . ,
B , ( A .
k -1)

(0)

. (

k -1)

(0)



( r + 1) - ( r + 1)

, - A -

, r a n k A = r . (4.33) . .
ij

( )



aij + bij A + B .
4.5. r = r a n k A k = N - r .



ij

( 0 ) = ij ( 0 ) = .. . =

( k -2)
ij

( 0 ) = 0, i, j = 1, N

.

(4.34)

. , k = 0 k = 1 (4.34) . k 2 . r n - 1 .
50


:

1.

§4



k 2 .

ij

( 0) = 0

. ,



ij

(0)



( N - 1)

- A .

r = N - k , N - k + 1, . . . , N .
j ( 0 ) = 0 . (4.31) i

ij ( ) =


1 l j N





ij

( )

,
l



ij

( )

l

(

)

i -

, j - l - aij + bij bi j .



ij

( 0)

,
l

bi j . , j ( 0 ) = 0 . i

( N - 2)

- ,

A . .

, 4.4. . . , (4.30) (4.34) , k = N - r , r = r a n k A . k r : k = N - r , (4.34) (4.30). ,

A=
:

1 1 0 0 2 3 1100 , B = 0 0 0 1 1 0 0 0 1 0

1 0 0 2 0 0 0 2 1 0 3 2

r = r a n k A = 2, k = n - r = 2, ( ) = 4 .
,

( 0 ) = ( 0 ) = ( 0 ) = ( 0 ) = 0 .
51


:

1.

§4

i, j = 1, 4 j ( 0 ) = 0 . i , i = j = 1

1 + 2

11

0

0
2

( ) =

0 0

1 + 2 1 + = 2 3 + 1 + 3 1 + 2
, 1 ( 0 ) 0 . 1

11

( 0 ) = 0 , 11 ( 0 ) = 0

, : k = N - r , r = r a n k A 4.5, . 4.3.4.1. () (). () ()

( A + B)

-1

=

-1

( A + B ) -1 ,

0

.

4.3.4.2. () ().

( A + B ) B = E - ( A + B ) A, 0 ,


-1

-1

( A + B) B E + ( A + B) A , 0 .
() (),

-1

-1

( A + B ) - 1 A E + ( A + B ) -1 B ,
­ () ().

0

4.3.4.3. () ().

0 < < -1 , = (T B ) ­ - T B ,


m=0



m (T B ) T ( B F + - E
m

)

52


:

1.

§4

. (4.7)

( A + B)

-1



-1

F

+

+

m=0





m (T B ) T ( B F + - E ) = O
m

(

-1

)

.

() () . 4.3.4.4. () ().

( A + B) B ( A + B)

-1

-1

B,

(4.9) (4.10). () () . 4.3.4.5 () (). (). ().
0

0 . -

(4.4). w R ( A ) R ( B Q ) . x k e r A y k er A , A x = w B Q y = w .

z = x - y



Qx = 0 ,

A y = 0 ,

( A + BQ ) z = 0

.


( A + B ) z = ( A + BQ ) z - B ( Q - E ) z


.

(4.35)

( Q - E ) z = ( E - A + A - E ) z = - A + A ( x - y ) = - x
z = - ( A + B

,

, (4.35),

) -1 B ( Q - E ) z = 2 ( A + B ) -1 B x

.

0 (), z 0 . z = x - y ,

53


:

1.

§4

y = 0 . , w = B Q y = 0 . (4.4) . () () . 4 .3 .5 . ( ) ( ) .

r = r an k A . d i m R ( A ) = r , d i m k e r A = N - r , d i m R ( Q ) = N - r .
() (). (4.4), (4.12)

di m R ( A) + d im R ( BQ ) = N .
,

d i m R ( B Q ) = N - r . .


di m R ( BQ ) < N - r .

z R ( Q ) = k er A , B z = 0 , ,

(

A + 0 B ) z = 0 , 0 ­ , (). -

, : d e t A + 0 B = 0 . (). () (). () (). (4.12). d i m R ( Q ) = N - r ,

(

)

di m R ( BQ ) N - r .
(4.12) ,

(4.36)

di m R ( A) + d im R ( BQ ) N .


di m R ( BQ ) N - di m R ( A) = N - r .
(4.36) (4.4).

54


:

1.

§4

0 . k 0 xk

,

d e t ( A + 0 B ) 0 . . d e t ( A + B ) = 0

(k )
N

-

,

(

A + k B ) xk = 0,
k

k = 1, 2, . . . .

x

xk = y k + z k , y k k er A , z k k er A ,

A z k + k B y k + k B z k = 0,

k = 1, 2, . . . . y

k

z

k

­

,

A z k = - k B y k - k B z k 0



k .

z k k er A

zk 0



k

z k / k . , , ,

z




k k

w0 ,

yk y0 ,

k ,

w0 k e r A,

y 0 k er A,

y0 = 1 .

w0 y0 A w0 + B y 0 = 0 . , ( A + B Q ) w0 + y

(

0

)

= A w0 + B y 0 = 0 ,
(4.37)

w0 + y 0 k e r ( A + B Q ) .
55


:

1.

§4

,

k e r ( A + B Q ) = {0} .

(4.38)

y k er A z k e r A ­ . ,

x = y + z , ( A + B Q ) x = A z + B Q y . (4.12)
R ( A + B Q ) = N , (4.38). (4.38) (4.37) , w0 + y 0 = 0 . w0 y
0

w0 = 0 y 0 = 0 .
0

y 0 : y

= 1 .
,

0

d e t ( A + 0 B ) 0 . () () .
() () . 4.3.6. () (). . 4.6. F G , G
0

:

{ }

G



M 1 = c o n s t 0 .



{(

F + G



)

-1

}

-

d e t F 0 . . .

{(


F + G



)

-1

}

0







:

(

F + G

)

-1

M 2 = con st .



d e t F 0 . . d e t F = 0 .

56


:

1.

§4

x0 , x
N

0

= 1 , , F x0 = 0 .



x0 = ( F + G



)(
-1

F + G



)

x0 = ( F + G



)

-1

G x 0 ,

0 .

,

1 = x0

(

F + G



)

-1

G



M 1M 2 0 0 .

d e t F 0 . . . d e t F 0 .
-1

{(

F + G
-1



)

-1

}

.







M2 F

(

)
)

-1

, G F

< 1




(

F + G

-1



=F

-1

(

E + G F

-1 -1

)

=F

-1

m=0

m ( - ) (

G F

-1 m

)

.

2 M

(

2 -1

F

-1

)

-1

,
-1

(

F + G



)

-1

F

-1

m =0





M
m

m 2

F

-1 m

F

m=0





1 =2 F 2m

.

. . ( [29], . 255). 4.7. K = , d e t K

K K

11 21

K1 2 K K 22

11

-

11

0 . d e t K 0 -

57


:

1.

§4



d e t K 2 1 K - 1 K1 2 - K
11

(


22

)







0 . , K ­ ,
-1

K

Y1 1 Y12 = , Y21 Y2 2



Y11 = K

-1 11

- K1-11K12Y0 K 21K1-11 , Y12 = K1-11K12Y0 ,

Y2 1 = Y0 K 2 1K1-11 , Y22 = -Y0 , Y0 = K 2 1K1-11 K1 2 - K
.

(

22

)

-1

.

K L= K


11 21

0 E

(

d e t L = d e t K11 0 ) .

E K = L 0
:

K

22

- K 21K1-11K12
12

K1-11K

d e t K = d e t K11 d e t K
d e t K
11

(

22

- K 2 1K1-11K

12

)

.

0

. . . 4.8. A , A1 1 ­ r , d e t A1 1 0 . r = r a n k A , A22 = A21 A11 A12 .
-1

58


:

1.

§4



A1-11 - A21 A1-11
(4.14) , A1

0 E

A11 A21

A12 E = A22 0

. A22 - A21 A1-11 A12

A1-11

4.9. A B
1

B1

1

­ r ,

r = r an k A , d e t A1 1 0 .


{

( A + B)

-1

}

0 < <

0

-

(4.15).

{

.

( A + B)

-1

}

.





:

( A + B)

-1

M 1 = co ns t,

0 < < 0 .

A + B

C11 C1 2 A + B = , C 2 1 C 22


Ci j C

ij

( ) = Ai j + Bi j ,
-1

i, j = 1, 2 .



( A + B)

, 4.7,

> 0

( A + B ) -1 =


D11 (

) D12 ( ) D2 1 ( ) D2 2 ( )

,
-1

(4.39)

D11 ( ) = C1-11 - C1-11C12 C 21C1-11C12 - C D12 ( ) = C1-11C12 C 21C1-11C12 - C
59

(

22

) )

C 21C1-11 ,
,

(

-1

22


:

1.

§4

D

21

( ) = ( C 21C1-11C12 - C
22

22

)

-1

C 21C1-11 ,

D

( ) = - ( C 21C1-11C12 - C

22

)

-1

.

(4.39) Di

{

j

( )},

i, j = 1, 2 .



D2 2 ( ) = [ F + G ( )


]

-1

,

F = A21 A1-11B11 A1-11 A12 - A21 A1-11B12 - B21 A1-11 A12 + B22 ,
G ( ) = - A21 ( A1 1 + B1 + A2 1 ( A11 + B1 +B
21 1

)

-1

B11 A1-11B1 1 A1-11 A1 2 +

1

)

-1

B1 1 A1-11B1 2 +
21

(

A11 + B1

1

)

-1

B1 1 A1-11 A1 2 - B

(

A1 1 + B1

1

)

-1

B1 2 .

4.6 (4.15). . . d e t A1 1 0 (4.15), 4.6,

{ {

D D

22 21

( )} ( )}

.







{ D11 ( )} { D12 ( )}



:

D12 ( ) = C1-11C12 D22 ( ) C1-11 C1

2

D2 2 ( ) ,
21

D21 ( ) = D22 ( ) C 21C1-11 D22 ( ) C

C1-11 .

60


:

1.

§4



{ {

,
-1

( A + B)
Di
j

( )} , i, j = 1, 2

}









. .

. () (). A B (4.14).

A1 1 A0 = A21
A + B

0 . E

E + A1-11B1 1 A + B = A0 B2 1 - A2 1 A1-11B1

(

1

)(

A1-11 A12 + A1-11 B12 , B22 - A21 A1-11B12

)



( ) = d et ( A + B ) =
E + A1-11B1 1 = d e t A0 d e t B21 - A21 A1-11B1

(

1

)(
1

A1-11 A12 + A1-11 B12 = B22 - A21 A1-11B12

)

E + A1-11B11 = d e t A11 d et B21 - A21 A1-11B1
k

A1-11 A12 + A1-11B12 . B22 - A21 A1-11B12

k = N - r . k - = 0



k

(

E 0 ) = k ! d et A11 d et B21 - A21 A1-11B1
k

1

. B22 - A21 A1-11B12
2

A1-11 A1

61


:

1.

§4



E -1 B21 - A21 A11 B1 E â 0

1

E = -1 B22 - A21 A1-11B12 B21 - A21 A11 B1
2

A1-11 A1

1

0 â E

, B22 - A21 A1-11B12 - B21 A1-11 A12 + A21 A1-11B11 A1-11 A12
2

A1-11 A1

:



k

( 0 ) = k ! k d et A11 â
2

â d e t B2 2 - A2 1 A1-11B1 2 - B21 A1-11 A12 + A2 1 A1-11B1 1 A1-11 A1
( A + B

(

)

.

(4.41)

{

)

-1

}

,

4.9 (4.15). (4.40) (4.13). () () . () (). (4.13). (4.40) (4.15), 4.9

{

( A + B)

-1

}

. () () .

4.2 . 4.4. 4.1. . (4.2) (4.1). , -, > 0 (4.2)

x . , A + B (4.2)
. -, x ­ .

{(

A + B

) -1 f

}

. f R ( A ) ­ , -

62


:

1.

§4



{(

A + B

) -1 A}

.

, (4.2) (4.1), (). 4.2 (). . . (). 4.2 ()-(), , (). (4.7)

( A + B ) - 1 f = -1F + f +

m =0





m (T B ) T ( B F + - E ) f .
m

f R ( A ) , P f = 0 , ,

F + f = F F

(

)

+

F f = F F

(

)

+

QB Pf = 0 .

> 0

x ( A + B

) -1 f = -

m =0





m (T B ) T f
m

(4.2). , x x0 .

x0 = -T f = A + f - F + B A + f
(4.1). , (4.2) (4.1). . 4.1 . 4.3. . (4.2) (4.1). 4.1 (), 4.9 ­ (4.15). .

63


:

1.

§5

. (4.15). (4.9)

{

( A + B)

-1

}



> 0 . (), 4.2 ­ (). 4.1 , (4.2) (4.1). . 4.3 .

5. . .

Ax = f


(5.1)

( A + E ) x = f
N , f
N

,

( 5 .2 )

A ­ , E ­ ­ , x
N

­ ,

­ .
(5.2) : 1) . . [108], A , f , 2) . . [48] . . [8], ( , ) A . A , (5.2) (5.1), , A C .

=

0





d e t ( A - E ) = 0
64


:

1.

§5

= 0 . , , = 0 , = 0 . = 0 , 1. =
0

,

1. 5.1. (5.2) (5.1), A . 5.1. . = 0 A k . U , A

J ( A ) .
J J ( A ) = U -1 A U = 0
0

0 , J1

J 0 ­ k , J 1 ­ , A . (5.1)

U -1 A Uy = g ,
y = U
-1

( 5 .4 )

x , g =U

-1

f . (5.1) ,

k g , g = g 0 , g

(

1

)

T

,

g 0 = 0 ­ k , g1 ­

N -k .











y:

y = ( y 0 , y1

)

T

, y0 ­ k , y1 ­
65


:

1.

§5

N - k . (5.4)

E k y 0 = 0, J 1 y1 = g1.
(5.2) :

(5.5)

(U

-1

AU + E y = g .

)

(5.6)



E k y 0 = 0, ( J 1 + E N - k ) y1 = g1.
J
1

(5.7)

­



k +1

, . . . , N .
0 < < min

{

k +1

, .. . ,

N

}

(5.8)

(5.7)

y = y

(

0,

, y1,



)

T

,

y

0,

= 0 , y1, y1, 0 0 , 0

y1,



(5.5). , (5.6) , (5.8), y , 0 .

x = Uy ­ (5.2). ,
(5.2) (5.1). . . (5.2) (5.1). , A . . A .
66


:

1.

§5

0 1 0 J = 1 0

,

k 2 .

A + E
1 . J + E = 1
J + E ,
1k

­ -

, k - .

= k,




1k

= ( - 1)

k +1

,

k -
k +1

(

J + E

)

-1



1k

/ = ( -1)

/ k . ,

J + E 4. (5.1).

{

(

)

-1

}

. ,

1 4.2, (5.2) A . . 5.1 .

C ­ N .

Cx = f


(5.9)

(C + E ) x = f
67

.

(5.10)


:

1.

§5

5.1. (5.10) (5.9). , . (5.10)

(

C + C0 ) x = f ,

(5.11)

C 0 ­ N . 5.2. (5.11) (5.9), C 0 . . . (5.11) (5.9). F , C C 0 .

F C F = d i a g ( 0 , 1 , . . . , F C 0 F = d i ag
m
0, m

N -1

)

,

(

0, 0

, 0,1 , . . . ,

0, N -1

)

,

, m = 0, N - 1 ­ .

(5.9) (5.10)

F C F y = F f , x = F y,


(5.12)

F C F + F C F y = F f , 0 x = F y.

(

)

(5.13)

, (5.12) , m = 0 , .

68


:

1.

§6

(5.9). y . (5.13) y .

m y
(5.12) (5.13). . , . 5.2 .

6. , § 4, . , 4.1 ­ 4.3, A B . A B , . ,

( A + B) x = f



x0 = A + f A x = f 0 , 0 B . 6.1. A B

R ( B ) = k e r A ,

(6.1) ( 6 .2 )

k e r B = R A .

()

A + B

( A + B ) -1 = - 1B + + A

+

.

(6.3)

. (6.1) , A B = B A = 0 . A + B = B + A = 0 , ,
69


:

1.

§6

(

-1B + + A + ) ( A + B ) = - 1B + A + A + A + B + B + A + B =
= A+ A + B + B .
( 6 .4 )

(6.2) ,

( k er B ) = ker A
( k e r B

,

(6.5)

)



­ k er B .

B + B ( k e r B

)



,

E - A + A ­ k er A . (6.5) B + B = E - A + A , A + A + B + B = E . (6.4) (6.3). . 6.1. 6.1. (4.2) (4.1) 0

x x 0 ,
x = ( A + B

(6.6)
+

) -1 f

­ (4.2), x0 = A f ­

(4.1). , (4.1) (4.2), , 0 . 6.1, , (4.2) (4.1) 0 . (6.1) (6.2) (4.2) (4.1). , (4.2) (4.1). .
70


:

1.

§6

6.2. (4.2) (4.1),

F +B = Q .

(6.7)

. . (4.2) (4.1). 4.1 4.2 (4.7). 0

x = ( A + B
+

) - 1 f x0 - F + B A + f

,

(6.8)

x0 = A f ­ (4.1).

x x0 0 . F + B A + f = 0 . f R ( A ) ­ , F + B A + A = 0 .

0 = F + BA + A = F + B E - E - A + A = F + B ( E - Q ) = = F + B - F + BQ = F + B - F + P BQ = F + B - F + F = F + B - Q .
(6.7) . . . (4.2) A + f - F + B A + f 0 . (6.7) F + B A + f = Q A + f = 0 . . . k er A k e r A , B . e1 , e2 , . . . , ek g1 , g 2 , . . . , g k k er A k e r A , .

((

))

B=

m =1



k

g m em ,

( 6 .9 )

71


:

1.

§6



e

m

­ -, -

em , m = 1, k
. (6.9)

Bx =

m =1

(

k

x, e

m

)

gm ,

( , ) ­ N . 6.3. B (6.9). 0

( A + B ) -1 =

-1

m =1



k

em g m + A + .

(6.10)

. :

+ A

-1

m =1



k

em g m + A + A +
-1

m =1



k

g m em = A + A +

+

m =1



k

g mem +

m =1



k

em g m A +

m =1 n =1 -1 k



k

k

e m g m g n en =

= A + A + A A +
k k

(

) ( m =1
+k

A g

m

)

em +

m =1

em (
m =1

A g

m

)



+

m =1 n =1

m (
e
k

g m , g n ) en = A + A +



k

em em .

(6.11)

,

m =1



em em = E - A + A .

72


:

1.

§7

z k er A . z =


k

k

n =1

z n en , z n ­

z e1 , e2 , . . . , ek .

m =1



k

em em z =

m =1 n =1



k

z n ( en , e

m

)

em =


m =1



k

z m em = z .

z ( k e r A

) = R ( A

)

.

y

N

, z = A y
em em z =

m =1



k

m =1

(

k

em , z ) em =

m =1

(
k

em , A y em =

)

m =1

(

k

A em , y ) em = 0 .

,

m =1



k

em e

m



k er A .

m =1



k

em em = E - A + A . (6.11)

(6.10). .

7. 7.1. . , , , . , . , . . , .

73


:

1.

§7

7.1. B ­ , 6.1. x0

Ax = f ,


f R( A

)

(7.1)

x0 = ( A + B

) -1 f

.

( 7 .2 )
k

e1 , . . ., ek g1 , . . ., g



k er A k e r A , ,
, , , , . 7.2. B (6.9)

f R ( A ) . x0 (7.1)

x0 = A +

m =1


+ 0

k

g m em

-1

f.

(7.3)

7.3. B (6.9). x

Ax = f ,


f N ,
-1

(7.4)

+ x0 = A +


m =1 k



k

g mem g ,

(7.5)

g= f -

m =1

(

f ,e

m

)

gm .

(7.6)

7.2. 7.1-7.3. 7.1 6.1 = 1 .

74


:

1.

§7

7.2. f R ( A ) , x
N

, A x = f . , = 1 (6.10),

A+
= A+ f +

m =1


k

k

g

e mm



-1

f = A+ +

m =1 k



k

em g m f =

m =1

(

f ,g

m

)

em = A + f + x , A g

m =1

(

A x , g

m

)

em =

= A+ f +
7.2 .

m =1

(
k

m

)

em = A + f .

7.3. f


N

f = f1 + f 0 , f1 R ( A ) , f 0 k e r A .

x1

N

1 , . . ., k , A x1 = f1 f 0 =


i =1

k

i g i .

g= f -

m =1

(

k

f ,g
k

m

)

g m = f1 + f 0 -

m =1

(

k i =1 k

k

f1 + f 0 , g

m

)

gm =

= f1 + f 0 - = f1 + f 0 -

m =1 k

(

A x1 , g

m

)

gm -

m =1 k


m =1 i =1

k

i gi , gm g m =
gi , g



m =1

(
k

x1 , A g

m

)

gm -

i (

m

)

gm =

= f1 + f 0 -

m =1



m g m = f1 + f 0 - f 0 = f1 R ( A ) .

, g R ( A ) . , (7.4) A x = g . 7.3 .

75


:

1.

§8

8. 8.1. . A ­ N , u N ­ .

u , A u , A 2u , . . . .

( 8 .1 )

, n : 1 n N ,

u, A u , . . . A

n -1

u

(8.2)

, A n u (8.2). , u N , (8.2) n = N . : A , u N , N (8.1)

u, Au, . .. A
.

N -1

u

(8.3)

u N (8.3) , (8.3) . . 8.1. (8.3) u N , , A . 8.1 8.4.

76


:

1.

§8

B = J ( A ) A . , A , Q :

QAQ


-1

=B,

(8.4)

1 B = J ( A) = 0

2

0 m

,
NâN

(8.5)

i A = i A = i i 0

1

i

0 1 i

, i = 1, . . . , m ,
k i âk i

(8.6)

k1 + k 2 + . . . + k m = N .
(8.2)

(8.7)

v, B v , . . . B
v = Q u .

n -1

v,

(8.8)

. 8.1. u N ­ . (8.2) (8.8). . Q

0v + 1B v + . . . +
77

n -1

B

n -1

v=


:

1.
- 1 n -1

§8

= 0Q u + 1Q A Q -1 Q u + . . . +

n -1

(

QAQ
A
n -1

)

Qu =

= Q 0u + 1 A u + . . . +

(

n -1

u

)

. 8.2. u N ­ . (8.2)

u , 1 0 u + A u , 2 0 u + 2 1 A u + A 2u , . . . ,



ij

n -1, 0

u +

n -1,1

Au + ... +

n -1, n - 2

A

n -2

u+A

n -1

u,

(8.9)

, 0 j < i n - 1 , ­ .

. . (8.2) ­ . (8.9),
ij

, 0 j < i n - 1 ­ .

:

x0u + x1 ( 10u + A u ) + x 2 20u + 21 A u + A 2u + . . . + x
n -1

(

)

(

n -1, 0

u +

n -1,1

Au + ... +

n -1, n - 2

A

n-2

u+ A

n -1

u.

)

(8.10)



( (
+x

x0 + 10 x1 + 20 x2 +...+

n -1, 0 n -1

x

)

u+

+ x1 + 21 x2 + 31 x3 +...+

n -1,1 n -1

x

)

Au + ... +
n -1

(

n -2

+

n -1, n - 2 n -1

x

)

A

n -2

u+x

n -1

A

u=0.

(8.2)

78


:

1.

§8

x0 + 10 x1 + 2 0 x2 + ... + n - 2, 0 x n - 2 + n -1, 0 xn -1 = 0, x1 + 2 1 x2 + ... + n - 2,1 x n - 2 + n -1,1 xn -1 = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xn - 2 + n -1, n - 2 xn -1 = 0, xn -1 = 0 .


x0 = x1 = ... = x

ij

n -1

=0

, 0 j < i n -1 .

, (8.10) . , (8.9) ­
ij

,

0 j < i n - 1 . ,

ij = 0 . ij = 0 (8.9) (8.2). . . (8.9) ( 8 .9 )
ij

, 0 j < i n - 1 . ,

ij







ij = 0 ,

0 j < i n - 1 .

(8.9) -

(8.2). (8.2) ­ . . . 8.2. . 1 , 2 , . . ., . N -
N

­

79


:

1.

§8

1 WN = 1 ... 1



N





N -1 N N -1 N -1

N -1

(8.11)

...

...

1

1N

-1

.

WN =

1 i N -1

(

i -

N

)

W

N -1

,

(8.12)



WN =

1 i < j N



(

i - j .
N

)

(8.13)

(8.13) , 1 , . . .,

­ ,

(8.11) . (8.12) (8.11), , -
N



, . :


WN =

N -1

-

N N



N -2

-

( N -1 ( N -2 (

- -

N N

) ) )



N -1 N -2



(N (N (

-1 -2

-

N N

-

) )



N -2 N -1 N -2 N -2



.

...

...
N

...

1 -
w

1 -

N

1



1 -

N

)

1N

-2

(8.12).
N

(, k )

, 1 k N , k â N
N -1

1, , 2 , . . .,

(

)

, -

.

w

N

(, k ) =
80

(8.14)


:

1.

§8

=

1... 01...



k -2 k -3



k -1 k -2

... ... ...



N -1 N -2

( ( ( 00... (
00... 00...

k - 2 )! k - 3)! k - 2 )! k - 4 )!

k -4

(k (k (k (k

- 1)! - 2 )! - 1)! - 3)!

k -3

( ( ( (

N - 1)! N - 2 )! N - 1)! N - 3)!

N -3

( k - 2 )!
0! 0

( k - 1)!
1! 0!



...

( k - 1)!

( N - k + 1)! ( N - 1)! ... ( N - k )!

( N - 1)!



N - k +1

N -k

.

1 , . . ., m , k1 , . . ., k m ­ ,

1 , . . ., m k1 + . . . + k m = N , 1 m N . WN k , . . ., k 1 m N -

wN ( m , k m ) w ( , k 1 , . . ., m WN = d e t N m -1 m-1 k , . . ., k 1 m wN ( 1 , k1 )

)

,

(8.15)

. ,

, , , W7 1 2 3 4 = 3, 2, 1, 1

81


:

1.
5 4 5 3 5 2 4 2

§8

1 1 1 =0

4 3 2



2 4 2 3 2 2 2



3 4 3 3 3 2 2 2



4 4 4 3 4 2 3 2





6 4 6 3 6 2 5 2

1

2

3

4

5

6

1 1 0 0 1 0

12
21 2

13
312

14
413

15
514

16
615

61 1 212

2 013 3 014

7. k1 = k 2 = . . . = k m = 1 , . , , , 1 , . . .,
m

.

N : 1) k 2 = . . . = k m = 1 ,

1 , 2 , . . ., m WN = k , 1, . . .,1 1 = ( 1 -

m

) (
k1 i =1

m -1

i -

m

)

W

N -1

1 , 2 , . . ., m k , 1, . . .,1 1

-1

,

(8.16)

2) k m m i n k1 , . . . , k

{

m -1

}

,

1 , 2 , . . ., m WN k , k , . . ., k 1 2 m

=

= 0!1!...( k m - 1)!

(
i =1

m -1

i -
82

m

)

ki k

m

W

N -k

m

1 , . . ., m -1 k , . . ., k 1 m -1

.

(8.17)


:

1.

§8

(8.16) (8.17) . (8.16) (8.17), . i , i = 1, m ­ , k i , i = 1, m ­ , N = k1 + . . . + k m .

1 , 2 , . . ., m WN k , k , . . ., k 1 2 m =

=


i =1

m

0!1!...( k i - 1)!

1 i < j m



(

i -

j

)

ki k

j

.

(8.18)

(8.18) (8.16) (8.17). 8.3. . (8.3) u
N

N -1

,


i =1

xi Ai u = 0 , xi = 0 ,

i = 0, N - 1 . , u
N N -1




i =1

xi Ai u = 0 -

. 8.1 (8.3) u
N

N -1

v, B v, B 2 v, . . . B

v

(8.19)

v N , B (8.4)-(8.7).
83


:

1.

§8

1 , . . .,

m

­ A ,
m

B . k1 , . . ., k



. , B , (8.15) (8.14). 8.2. B , ,

1 , . . ., m WN . k , . . ., k 1 m
8.2 . 8.3.

x0 E + x1C + x2C 2 + . . . + x
C = d i a g c1 , c2 , . . . , c

m -1

C

m -1

= 0,

(8.20)

(

m

)

, ,

, c1 , c2 , . . . , cm . . (8.20)

x x . x

0 0

+ c1 x1 + c12 x 2 + . . . + c1m -1 x
2 + c 2 x1 + c 2 x2 + . . . + c

m -1

= 0, = 0, = 0.
(8.21)

m -1 x m -1 2 m -1 xm -1 m

.....................
0 2 + cm x1 + cm x 2 + . . . + c

(8.21) , , (8.21) . (8.21) -

84


:

1.

§8

. , c1 , c2 , . . . , c . .

m

0 d 1 D= 1 d 0
8.4.



.
mâ m

x0 E + x1D + x 2 D 2 + . . . + x

m -1

D

m -1

=0

(8.22)

. . (8.22)

( d ) ( d ) ( d ) ( m -1) ( d ) , , , . . ., , 0! 1! 2! ( m - 1)!


( d ) = x0 + x1d + x 2 d 2 + . . . + x

m -1

d

m -1

.

(8.22)

x0 + x1d + x 2 d 2 + . . . . . . . . . x1 + 2 x2 d + . . . . . . . + ( m 1 x2 + . . . + ( m - 1) ( m 2 . . . . . . . . . . . . . . . . . . . . . .

. +x - 1) x

m -1

d d d

m -1 m-2 m -3

= 0, = 0, = 0,
(8.23)

m -1

- 1) x

m -1

......... xm -1 = 0 .

xi , , (8.23) . .

85


:

1.

§8

8.2. . B , 1 , . . .,
m

. ,

(8.18), . . . , (8.18) , 1 , . . ., 1 , . . .,
m

-

. 8.3 8.4 , m



, . , B . . . 8.2 . 8.4. 8.1. . (8.3) . A . , (8.3) 8.1 (8.19) v N . ,
N -1


i =0

xi B i v xi = 0, i = 0, N - 1 .
N -1




i =0

xi B i v = 0

xi , i = 0, N - 1 . , ( W N

1 , . . . , m , k1 , . . . , k m

86


:

1.

§8

A ) . 8.2 , B ( A ) . . . A . u N , (8.3) N . , 8.2, A ( B ), , W N

1 , . . . , m , k1 , . . . , k m 1 , . . . , m 0 . k1 , . . . , k m

A , : W N

(8.3)

u N , (8.19) ­ . xi , i = 0, N - 1 ,
N -1 N -1


i =0

xi B i v :


i =0

xi B i v = 0 . ,

N -1


i =0

xi B i v = 0 . , -

(

1 , . . . , m WN , A ) k , . .. , k 1 m
. , . , u N , (8.3) . . 8.1 . 8.1.
87


:

1.

§8

(8.4)-(8.7). u
0i





k i ,
T

u 0i = ( 0, ..., 0, 1) , i = 1, m .
:

(8.24)

(

Ai -i E

k

i

)

r

u 0i = e
k

k i k -r i

()



0 r ki - 1 ,
rk , i

(8.25) (8.26)

(
E
k i

Ai -i E

)

r

u 0i = 0

­ k i ,

el

( ki ) = (

0, ..., 0, 1, 0, ..., 0

)

T

­ k i , l - . u 0 = u 0 1 , u 0 2 , . . ., u

(

0m

)

T

, u

0i

­ -

k i (8.24). ,

u0 , Bu 0 , ... B

N -1

u

0

(8.27)

.

0u 0 + 1 B u 0 + ... + 1 B
(2)

n -1

u0 = 0 .

(8.28)

0u 01 + 1 A1 u 01 +...+ n-1 A1n-1u 01 =0, n 0u 02 + 1 A2 u 02 +...+ n-1 A2 -1u 02 =0, . . . . . . . . . . . . . . . . . . . . . . . . . . n -1 0u 0m +1 Amu 0m +...+ n -1 Am u 0 m =0 .
88

(8.29)


:

1.

§8

, , 1 : 1 = 0 . i 0

2 i m , , Aip 2 i m

p = 1, 2, . . . , ,

d e t Aip 0



2im,

p = 1, 2, . . . , .

(8.30)

2

u01 ,

A1u 0 1 , ...,

A1

k -1 1u

01

(8.31)

,

u01 ,

(

A1 -1E

k 1

)

u 0 1 , ...,

(

A1 -1E

k 1

)

k -1 1

u 01 .

(8.32)

(8.25) (8.32) . , (8.31) . (8.25) (8.26) (8.29)

0e
e1
( k1 )

(k ) 1 k 1

+ 1e

(k ) 1 k -1 1

+...+

k -1 1 1

e

(k ) 1

=0.

(8.33)

, . .. , e

( k1 ) k1

- , -

(8.33)

0 =0, 1 =0, ...,
k +1 1u

k -1 1

=0 .

(8.34)

, (8.29)

k A21u 02 +
1

k

k +1 2 1

A

02

+...+

A n-1u 02 n -1 2
n - k -1 1u

=0.

(8.35)

(8.30) , (8.35)

k u 02 +
1

k +1 2 02 1

Au

+...+
89

n -1 2

A

02

= 0.


:

1.

§9

, (8.29), :

k =0,
1

k +1 1

=0, ...,

k + k -1 12

=0 .

, :



k +k 12

=0,

k + k +1 12

=0, ...,

k + k +. . .+ k -1 12 m

=0 .

k1 + k 2 + ... + k m = n ( (8.7)), (8.28)

0 =0, 1 =0, ...,

n -1

=0 .
-1

(8.27). 8.1 , (8.1) u = Q u 0 , . Q ­ , A B ( (8.4)). 8.1 .

9.
N

-

e1 , e2 , . . ., e N . A . : 9.1. A . µ1 , µ 2 , . . ., µ N C , µ1 , µ 2 , . . ., µ A + C .
N



90


:

1.

§9

. . 8.1 u
N

,

u , A u , A 2u , . . . , A

N -1

u
N -1

(9.1)

N . A N u (9.1)

A N u = -a N u - a
a1 , . . ., a
N

N -1

A u - . . . - a1 A

u,

(9.2)

­ . (9.1) -



A=

0...0 1...0
0...1

U ­
-1

-a N -1 . -a1 e1 , e2 , . . ., e
N

-a

(9.3)

N

(9.1),

, , A A

A = UAU

.
N

µ1 , µ 2 , . . ., µ

­ .

c1 , c 2 , . . ., c N , cm = ( - 1)
m



µ k µ k . . .µ k ,
1 2 m

m = 1, N ,

(9.4)



k1 , k 2 , . . ., k m : 1 k1 < k 2 < . . . < k m N .
,



m

(

cm = ( - 1)
m

z1 , z 2 , . . ., z

N

)

m

(

µ1 , µ 2 , . . . , µ

N

)

,

m = 1, N ,



m z1 , z 2 , . . ., z N .

91


:

1.

§9



a1 , a 2 , . . ., a

N

0 0 C= 0 ­

...0 ...0 ...0


a N -1 - c N -1 , (9.5) a1 - c1 (9.2), c1 , c 2 , . . ., c N ­
N

aN - c

(9.4). , C . (9.3) (9.5)

A+C =

1 ... 0 -c N -1 . 0 ... 1 -c1 0...0
N

-c

(9.6)

, A + C

N + c1

N -1

+ . .. + c
N

N -1

+ cN = 0 .

(9.7)

c1 , c 2 , . . ., c

( (9.4)) [45]

, (9.7) µ1 , µ 2 , . . ., µ N . , A + C µ1 , µ 2 , . . ., µ N . :

A =U

-1

AU .



C = U -1 C U . C , C ­
. A + C A + C :

A+C =U
µ1 , µ 2 , . . ., µ
N

-1

(

A+C U ,

)



A+C .
.
92


:

1. § § 0 19

10. 10.1. ().

H

­

,

A : H H B : H H ­ . , ( ) ( ). () Ax = f ,
(10.1)

f H ­ , x H ­ . (10.1)

( A + B) x = f

,

(10.2)

­ . , (10.2) (10.1), : ) f H (10.2)

x = x



(f)

0 = : 0 <

{

0

}

, 0 ­ (10.2)

; ) f R ( A

)

x = x
0



(f)

0 : x x0 = x (10.1): A x0 = f .

(f)

, x0

) ,

0



A + B , M > 0
,

( A + B)

-1

M ,

0 .

(10.3)

93


:

1.

§9

A , M M 0 = c o n s t , A ,

M 0 .
)
0

,







( A + B) A
A A0 = A .

-1

,



A0 : H H 0 ,
, A , B 0 =
0

( A, B ) > 0

,

0 < < 0 A + B . x0 ­ (10.1),

x - x0 = O ( ) .

(10.4)

, A ­ , (10.2) (10.1) B . A ­ , , -, (10.1) f H , -, (10.1) (10.2), (10.4). ( A, B

)

A B , (10.2) (10.1). 10.2. . . . 1. R ( A ) R A

( )

-

A , ,

94


:

1.

§9

,

R A k e r A

()



R ( A ) k e r A ;

R ( A ) R A

( )



, A , A , R ( A ) R A

( )

-

, , . 2. x


(f)



(10.2) 0 R0 R ( A ) , , . 3. A B , (10.2) (10.1), A B . ,

H = l 2 A ,

A x = ( 0, x1 , x2 , . . .) ,



x = ( x1 , x 2 , . . .) .



B : l 2 l 2 , (10.2)
f l 2 . , : y l 2 , y = y1 , y 2 , y3 , . . . , y = y1 , y

(

)

(



)

, y = y 2 , y3 , . . . .

(

)

A A x = ( 0, x ) . B ­ -

95


:

1.

§9

. B x = B1 x, B2 x , B1 x ­ B x ,

(

)

B2 x ­ . (10.2)

B1 x = f1 , ( E + B2 ) x = f 2 .

0

B

2

< 1 ,

x = E + B

(

2

)

-1

f 2 .

x f1 f 2 :

B1 ( E + B
f = ( f1 , f

2

)

-1

f 2 = f1 .


2

)

l 2 . , , f = (1, 0, 0, 0, . . .) l 2 ,

f1 = 1, f 2 = ( 0, 0, 0, . . .) , . 4. A B , (10.2) f l
2



0 , , ,
(10.2) 0 (10.1), f R ( A ) . ,

Ax = f 0 ,

1 1 1 f 0 = , 0, , 0, , 0, . . . l 2 , 4 6 2

A x = x2 , 0, x4 , 0, x6 , 0, . . . .

(

)

1 1 1 x0 = 0, , 0, , 0, , 0, .. . l 2 . 4 6 2
B .

( A + B ) x = ( x1 + x2 , x2 ,.. ., x
96

2 n -1

+ x 2 n , x 2 n , . . .) .


:

1.

§ 10

( A + B ) x = f 0 0 :

x



( f0 )

= ( A + B
-1

) -1 f 0 =
f

f f f f = 1 - 2 , 2 , . . ., 2 n 2

-



2n 2

,

f

,. .. =
2n

1 1 1 1 = , 0, , 0, , 0, . . ., , 0, . . . . 4 6 2n 2
, x


( f0 )

(10.2) -

0 x0 + y (10.1), y ­ k er A . 10.3. . : A + ­ A [39, 104]; Q ­ k er A ­ A ; P ­

ker A



­

A ;

F = P BQ ; A;


F0 = F

ker A

­





F






0

T = F + B - E A+ ;

(

)

T0 = B F + - E ;

0 = : 0 < z

{

}

,

0 ­ .
10.1. : (1) (10.2) (10.1); (2)

( A + B)

-1

M = con st,

0 .


(10.5)

(3) A

F0 : k e r A k e r A
; 97

(10.6)


:

1.

§ 10

(4)

( A + B ) B M 1 = co nst ,
(5)

-1

0 ;

(10.7)

( A + B ) -1 A M 2 = c o n s t ,
(6)

0 ;

(10.8)

B( A + B)

-1

M 3 = co n s t ,

0 ;

(10.9)

(7)

A( A + B

)

-1

M 4 = con st ,

0 ;

(10.10)

(8) A

F0 : k er A k e r A
;

(10.11)

(9) A F

F + F = Q,

F = PB Q ;

(10.12)

(10) A F

F F + = P,

F = P BQ ;

(10.13)

(11) A F

( A + B ) -1 = - 1F + +
: 0 < 0 < T B : H H ;
-1

m =0





m (T B ) T ( B F + - E ) ,
m

(10.14)

, = (T B

)

­ -

98


:

1.

§ 10

(12)

k e r A k e r B = {0} , R ( A) R ( BQ ) = H ;
(13)

(10.15) (10.16)

k e r A k e r B = {0} ,

(10.17) (10.18)

R A R B P = H .
10.1 10.5.

()

(

)

(10.1) (10.2)

A y = g ,

(10.19) (10.20)

(

A + B



)

y=g.

. 10.1, , : 10.2. , (10.20) (10.19), , (10.2) (10.1). 10.2 10.6. 10.1 A A . 10.3. (10.2) (10.1). A A . 10.3 10.6.

99


:

1.

§ 10

, , 10.3 B , (10.2) (10.1). 10.4. A : H H d i m k e r A = d i m k e r A , k er A ­ , A A , k er A ­ . B : H H , (10.2) (10.1). 10.4 10.6. 10.4. . [43, . 198], . 10.1.

{ n}

,

H , , : 1) .

n C = co n s t ,
2)
n

n = 1, 2, ... ;
x -

( x) ( x)

X , H . . , x0 X ,

k ( x0 ) ( x 0 ) ,

k .
k

x H ­ . ,

( x) ( x)



k . , > 0 x0 , 100


:

1.

§ 10







X,





x0 - x < .

k ( x) - ( x)
k ( x) -k ( x

k 0

)

+

k

( x0 ) ( x0 )

- (x

0

) ) )

+ ( x0 ) - ( x ) + x0 - x < + C .
k .
,

x - x0 +
< C +

k

- (x

0

k

( x0 )
0

- ( x

0







k ( x0 ) ( x

)



k ( x ) - ( x ) 0 k x H .
10.1 . 10.2.

{ yn }

­

H :

y n C = co n s t ,

n = 1, 2, ... .



{ y ( )}
k n

-

, y 0 H . . Y X = y1 , y 2 , . . . . Y Y H , , . n ,

{

}

n ( x ) = ( yn , x ) , x H ; n = yn , n .

n

C ,

101


:

1.

§ 10

1 ( y1 ) ,

2

( y1 )

, ...

n

( y1 )

, ...

.
( ( 1(1) , 21) , . . . n1) , . . . ,

{ n }





1(1) ( y1 ) ,

(1) 2

( y1 )

( , . . . n1) ( y1 ) , . . .

. ,

{ ( )}
1 n



1( 2) , 1(
)

(2 2

)

( , . . . n2 ) , . . . ,


2

( y2 )

( ( , 22) ( y 2 ) , . . . n2) ( y 2 ) , . . . .

,
( ( 1(1) , 21) , . . . n1) , . . .

1( 2) ,

(2 2

)

( , . . . n2 ) , . . .

..............
(k ( ), n )

{}

y1 , y 2 , . . . , y k , . . . . , «»

1(1) ,

(2 2

)

( , . . . nn ) , . . . ,

(n n ) ,

{}

102


:

1.

§ 10

1(1) ( y n ) ,

(2 2

)

( yn )

, ...

(n n . 1 n ) ( x

{

)}

-

x Y :



(n n

)

( x ) (0) ( x )

,

x Y .

y 0 Y ,
n

( l i m nn ) ( x ) = y 0 , x

(

)



x Y .
.
0





zH

­



z = x + x0 , x Y , x0 Y , y 0 , x

(

)



= 0 ,

(
,
n

y0, x = y0, x + y0, x

)(

)(

0

)=(

y0, z .

)

( li m nn ) ( z ) = y 0 , z

(

)



zH .

10.2 . 10.3.

F ( ) = ( A + B

)

-1

0 = : 0 < . .
0

{

0

}

, 0 > 0 ­ -

0 .

F ( ) - F (

0

)=(

0 - ) F ( 0 ) B F ( )
-1



F ( ) = E + ( - 0 ) F ( 0 ) B
103

F (

0

)

.


:

1.

§ 10

-

0

F ( 0 ) B < 1 , E + ( - 0 ) F ( 0 ) B
-1

[43] - = 0 . 10.3 .

(

0

)

, -

F ( ) -

10.5. 10.1. (1) (2). (10.2) (10.1). M > 0 , (10.5). , ) , x H M x > 0 ,

( A + B ) -1 A x M x ,

0 .

(10.21)

(10.21), ( -, [44, 97]),

M 1 > 0 ,

( A + B ) -1 A M 1 ,

-1

0 .

(10.22)

( A + B) B = E - ( A + B) A ,


-1

( A + B ) B M 1 + 1,

-1

0 .

104


:

1.

§ 10

0

0

­

0 = 0 .
0

M 0 = ( A + 0 B

)

-1

. , M 0 < . 0 <



( A + B) =
= ( A + B


-1

) - 1 A + 0 ( A + B ) - 1 B ( A + 0 B
-1

)

-1

,

( A + B)

M1 +
0

(

0

(

M 1 + 1) M 0

)



(

2 M 1 + 1) M 0 M .

(1) (2) . (2) (3). A .
0





F ( ) = ( A + B

0

) -1 ,

0 .

(10.23)

1 ­ 0
0

r0 r1 ( 0 < r0 < r1 <

)

, .

r0 < < r1 . ,

F ( ) = F1 ( ) - F0 ( ) ,


0 ,
1 i 2

(10.24)

F1 ( ) =

1 i 2




1

F (z) d z, z -

F0 ( ) =




0

F (z) dz . z -

105


:
i

1.

§ 10

F0 ( ) . z = r0 e ,

F0 ( ) =

1 i 2

2


0

F r0e
i

(

i

)

r0e - F r0e
i

i r0e i d =

=
r0 e
i

2

r0

2


0

(

i

)

r0e -

e i d .

(10.25)

- - r0 > 0 , , (10.25),

F0 ( )
r0
2

2

r0

2


0

F r0e
i

(

i

)

r0e -

e

i

d

2


0

r rM M 2 M d = 0 =0 . 2 - r0 - r0 - r0

(10.26)

r0 : 0 < r0 < . , (10.26) r0 0 ,

F0 ( ) 0



r0 0 .

(10.27)

, (10.24) (10.27)

F ( ) =

1 i 2




1

F (z) dz, z -

0 < < r1 .

, = 0 . ,

0 ,
C
-1

li m F ( ) =
0

F ( z) 1 1 li m dz = 0 z - i 2 i 2
1




1

F ( z) dz = z
,

=

1 i 2




1

z ( A + zB z

)

-1

dz =

1 i 2



(
1

A + zB

) -1 d z

106


:

1.

§ 10



C

-1

=

1 i 2



(
1

A + zB

) -1 d z

.

(10.28)

F ( = 0 , F ( 0 ) = C
-1

)

-

. (10.28)

C -1 f = l i m ( A + B
0

) -1 f =

1 i 2



(
1

A + zB

) -1 f d z

f H . f R ( A ) . (10.22)

C -1 f = l i m ( A + B
0

) -1 f = 0

.

(10.29)

, f R ( A )

x0 H : A x0 = f . (10.22)
li m ( A + B
0

lm ) -1 f = i0 ( A + B ) -1 A x0 = 0

.

. (10.29) , f R ( A ) . , > 0 ­ . : 1) f R ( A ) ,

f - f <
2)


2M

;


> 0 , : 0 < <



( A + B)

-1

f



<


2

,

(10.29). 107


:

1.

§ 10

,

( A + B)

-1

f ( A + B

)

-1

f-f
+



+

+ ( A + B

) -1 f < M


2M



2

= .

, 0 < < mi n 0 ,

{

}



( A + B)

-1

f < . > 0 ­ ,

(10.29) f R ( A ) . , ,

x = ( A + B

) f , f R ( A)

-1

(10.2) 0 .

f H

( ) =
F ( (

F ( ) f - F (0) f , -0

0 ,
-1

)

(10.23), F ( 0 ) = C
0

­ (10.28). -

)



[33, . 274]

0 f H . ( ) M f .
[44]

M 0 > 0 , ( ) M

0

f.

f R ( A ) . F ( 0 ) f = 0

( A + B) ( ) =
,

-1

f

= x ,

0 .

x



= ( ) M

0

f,

0 .

108


:

1.

§ 10

x



­ (10.2), A x + B x f , 0 .

n 0 ­ .

{x }

n

, -

x0 . ,

{x }

n

x0 . ,

( )

A x + n B x
n



f,
n

n , A x0 = f . ,

f R ( A ) . , , f R ( A
A .

)

f R ( A ) ,

F = P B Q : k e r A k e r A .


H = ( E - Q ) H QH H = ( E - P ) H PH .

(10.30)

(10.30) AB

A A= 1 0


1

0 , 0

B1 1 B= B2 1

B12 , B2 2

A1 1 = ( E - P ) A ( E - Q ) ,

B1 1 = ( E - P ) B ( E - Q ) ,
22

B1 2 = ( E - P ) B Q, B2 1 = P B ( E - Q ) , B

= PBQ .

109


:

1.

§ 10

x f

x = ( x1 , x

2

) )

: :

x1 ( E - Q ) H , f1 ( E - P ) H ,

x2 QH , f 2 PH .

f = ( f1 , f

2

(10.2)

( A11 + B11 ) x1 + B1 2 x 2 = f1 , B21 x1 + B22 x2 = f 2 .
(10.31) f1 , f x1, , x
2,

(10.31)

2

-

0 . A : H H

,

A1 1 : ( E - Q ) H ( E - P ) H


(10.32)

A1-11 : ( E - P ) H ( E - Q ) H .
0 A1-11 B11 < 1 ,

A1 1 + B1 1 : ( E - Q ) H ( E - P ) H


(

A11 + B1

1

)

-1

: ( E - P) H ( E - Q) H .


(10.31) f1 = 0 x1,

x1, = - ( A11 + B1

1

)

-1

B12 x

2,

.

(10.31)

B

22

(

x

2,

)

- B

21

(

A11 + B1
110

1

)

-1

B12 x

(

2,

)

= f2 .

(10.33)


:

1.

§ 10

(10.5)

{ x }
2 ,
2,

, n 0 n .

0 : x
y
n

2,

C . y n = n x

n

C , n = 1, 2, . . . . 10.2

{ yn }



{y }
n
k

.

y 0 . (10.33)

B

21

(

A11 + B1

1

)

-1

B12 x

(

2,

)

0
n



0.

, (10.33) = k ,

k

B2 2 y 0 = f 2 .

(10.34)

f 2 P H ­ , (10.34) , B
22

: Q H P H «» , -

, : R B

(

22

)

= PH .

, f 2 P H (10.34) ,

ker B
, x
2,0

22

= {0} .

(10.35)

k e r B22 . (10.32) -

, x1,0 ( E - Q ) H ,

A1 1 x1,0 = B12 x

2,0

.

(10.36)

111


:

1.

§ 10

(10.31) x1 = 0 , x2 = x

2,0

(10.36)

( A11 + B11 ) 0 + B12 x 2,0 = A11 x1,0 , B21 0 + B22 x 2,0 = 0 .
:

(10.37)

x0 = 0, x

(

2,0

)

H ,

y 0 = x1,0 , 0 H ,

(

)

f 0 = A11 x1,0 , 0 H .

(

)

(10.37) ( A + B ) x0 , A y 0 . (10.37) , 0 ,

( A + B ) x0 = A y ) -1 A y

0

, -

x0 = ( A + B

0

.



{(

A + B

) -1 A}

,

0
x0 = ( A + B
, x (10.35). , B
22 2,0

) -1 A y 0


( A + B ) -1 A y 0 0

0.

= 0 . -

: QH P H


. Q H = k e r A , P H = k e r A


B22 = P B Q = F0 ,





F0 = B22 : k er A k e r A .
(2) (3) .

112


:

1.

§ 10

(3) (1). 0 > 0 , : 0 < 0

A + B : H H

(10.38)

. , (2) (3), (10.2) «»

( A11 + B11 ) x1 + B1 2 x 2 = f1 , B21 x1 + B22 x2 = f 2 .
B B
-1 22
22

(10.39)

= P B Q = F0 -

: k e r A k e r A . (10.39)
-1 -1 x2 = - B22 B21 x1 + -1B22 f 2 .

(10.40)



(

-1 A1 1 + B1 1 - B12 B2 2 B

21

)

-1 x1 = f1 - B1 2 B2 2 f 2 .

(10.41)

A

A1 1 : ( E - Q ) H ( E - P ) H .

-1 0 < 0 , 0 A1-11 B11 - B1 2 B22 B

(

21

)

<1,

(10.42)



-1 A1 1 + B11 - B1 2 B2 2 B

(

21

)

: ( E - Q) H ( E - P) H

. (10.42). (10.41)
-1 x1, = A11 + B1 1 - B12 B22 B

(

(

21

)) (
-1

-1 f1 - B1 2 B22 f

2

)

.

113


:

1.

§ 10

(10.40)

x

2,

-1 -1 = - B2 2 B2 1 x1, + -1B22 f 2 .

(10.38) 0 < 0 . , (10.42) (10.2) f H , ). ).

f = ( f1 , f

2

)

A ,

f = ( f1 , 0 ) . (10.39)
-1 x = A + B - B B -1B f1 , 11 11 1 2 22 21 1, x = - B -1B A + B - B B - 1B 22 2 1 11 11 1 2 22 2,

(

(

))

(

(

21

))

-1

(10.43)

f1.

(10.43) 0:
0 1, 0 2 ,

li m x

= A1-11 f1 x1,0 ,
2,0

li m x

-1 = - B22 B21 A1-11 f1 x

.

, x0 = x1,0 , x (10.1):

(

2,0

)

H

A A x0 = 1 0

1

0 0

x1,0 A A -1 f f1 = 11 11 1 = x2,0 0 0

= f .

(3) (1) . (2) (4). (10.5),

( A + B) B ( A + B)
114

-1

-1

B M B .


:

1.

§ 10

M 1 = M B , (10.7). (2) (4) . (4) (5). (4) (5).

( A + B ) -1 A = E - ( A + B ) -1 B
(10.7),

,

( A + B ) -1 A 1 + M

1

.

M 2 = 1 + M 1 , (10.8) (5) (4).

( A + B) B = E - ( A + B) A ,
(10.8),

-1

-1

( A + B) B 1+ M 2 .
M 1 = 1 + M 2 , (10.7) (4) (5) . (5) (2). 0
0

-1



0 = 0 . M 0 = ( A + 0 B

-1 -1

)

-1

(, M 0 < ). -

( A + B ) = ( A + B ) A + 0 ( A + B ) B ( A + 0 B
-1

)

-1

,

(10.7) (10.8),

( A + B)

-1

M 2 + 0 M 1 M 0

(

)

0

(

M 2 + M1 ) M 0 .

115


:

1.

§ 10

M =

0

(

M 2 + M 1 ) M 0 , -

(10.5). (5) (2) . (2) (6). (10.5),

B( A + B)

-1

B ( A + B

)

-1

B M.

M 3 = B M , (10.7). (2) (6) . (6) (7). (6) (7).

A( A + B
(10.9),

) -1 = E - B ( A + B )
.

-1

,

( A + B ) -1 A 1 + M

3

M 4 = 1 + M 3 , (10.10) (7) (6).

B ( A + B ) = E - A( A + B ) ,
(10.10),

-1

-1

B( A + B)

-1

1+ M 4 .

M 3 = 1 + M 4 , (10.9) (6) (7) . (7) (2). 0
0



0 = 0 . M 0 = ( A + 0 B


)

-1

(, M 0 < ). -

116


:

1.

§ 10

( A + B ) = ( A + 0 B
-1

)

-1

A ( A + B

) - 1 + 0 B ( A + B ) - 1
0M

,

(10.9) (10.10),

( A + B)
M =
0

-1

M

0

(

M 4 + 0 M

3

)

0

(

M4 + M

3

)

.

(

M4 + M

3

)

M 0 , -

(10.5). (7) (2) . (2) (8).

( A + B

-1

)

= ( A + B

)

-1

M,

0 .

(2) (3) (2) (8). (8) (2). A A , F0 F0 . (3) (2) (8) (2). (2) (8) . (3) (9), (3) (10). A (3). F


F0 : k er A k e r A . ,
F0
-1

(

ker A



)

F0 , F = F0 .
+

-1

+

-1

­ , F
+

­ . F F .

-

117


:

1.

§ 10

(10.12) (10.13) :

k er F + F = ker Q



k e r F F + = k er P .

F = P B Q , k e r Q k e r F + F .
+




+


+



k e r F F k e r Q . F F x = 0 . F = F Q , F F Q x = 0 .
k er A F F0 , , F = F0 .
+ -1

0 = F + F Qx = F0-1F0Qx = Qx .
k e r F + F k e r Q . (3) (9) .
+

F + = F F

(

)

+

F





F = QB P ,





k er P ker FF

. -

k e r F F + k e r P . F F + y = 0 .

k er A F F + = F0-1 .



F0 , ,

0 = F F + P y = F0 F0-1P y = P y .
k e r F F + k e r P . (3) (10) . (9) (3), (10) (3). A (9) (10). F F0 . F0 . (10.12) (10.13) ,
+

118


:

1.

§ 10

F0

+

k e r A .
-1

, F0
-1 +

-

F0 = F0 . , F0 . (9) (3) (10) (3) . (10) (11). A F (10). , , (10.14), A + B . A + B (10.14):

( A + B ) -1 F + +




m =0





m (T B ) T ( B F + - E ) =
m



= -1 A F + + A T B F + - E + B F

((

)

+

)
)

+
.

+

m =1

m (



AT + E ) ( BT

) m ( BF + - E

, E . A Q = 0 ,

AF + = AF



(

FF

+

)

= AQB P FF

(

+

)

=0.

T = F + B - E A + A F + = 0 , (10.13)

(

)

A T B F + - E + BF + = A F + B - E A
= A F + BA + - AA = - AA
+

(

)

(

) +( ) (

B F + - E + BF + =

)

(

+

)(

BF + - E + B F + =
+

(

B F + - E + BF + = AA + + E - AA

)

)

BF + =

= - P + E + P BF + = - P + E + PB QF + = - P + E + FF + = E ;
119


:

1.

§ 10

( A T + E ) BT = ( A ( F + B - E ) A + + E ) BT =
= A F + B A + - A A + + E BT = - AA + + E BT = = PB T = P B F + B - E A + = PBF + BA + - P BA + =

(

)

(

)

(

)

= PB QF + B A + - P BA + = FF + B A + - P BA + = = FF + - P B A + = 0 .
(10) (11) . (11) (2). A F A + F + . (10.14) . ( A + B

(

)

{

)

-1

}

(10.14).

(11) (2) . (3) (12).

k e r A k e r B = {0} , R ( A ) R ( B Q ) = {0} , R ( A) + R ( BQ ) = H ,
R ( A ) R ( B Q ) .

(10.44) (10.45) (10.46)

(10.46) -

(10.44). z k e r A k e r B .

z k er A Q z = z . z k er B Bz = 0 P BQz = 0 F z = 0 .


F : k er A k er A , z = 0 .
(10.45). w R ( A ) R ( B Q ) . w R ( A ) x k e r A , A x = w . 120


:

1.

§ 10

w R ( B Q

)


y k er A ,

B y = w . z = x - y .

( A + B ) z = 2 Bx

2

.





( 3 ) ( 2 )



0 = 0 ( A, B ) > 0 , A + B

.

0 = { : 0 <


0

}

.



0 ,



{

z =

( A + B ) -1 B x





( A + B )1 B}
z


(. (1) (2))



= ( A + B

) -1 B x 0



0.

, z = x - y - y 0 . y = 0 . , w = B y = 0 . (10.46). f H ­ .

P f k er A .

F = P B Q : k er A k e r A y k er A : F y = P B Qy = P B y = P f . g = f - By .
(10.47) P g = P f - P B y = 0 , g k e r P = R ( A ) . A , g R ( A ) . x k e r A : A x = g . (10.47) A x = f - B y ,

A x + B y = f . (10.46).
, (. [96])

H 1 + H 2 = { x1 + x 2 : x1 H 1 , x 2 H

2

}

H 1 H H 2 H H 1 H 2 , : 121


:

1.

§ 10

1) H 1 H 2 ­ ; 2) H 1 H 2 = {0} ; 3) H 1 + H 2 = H . , (10.45) (10.46) R ( A ) R ( B Q ) = H . R ( A

)



A , R ( B Q ) F : k er A k er A . , F : k er A k er A M > 0 , F x M x x k er A .

Fx
,

2

= P BQx + ( E - P ) BQx , x k er A
2 2

Bx

2

PBQx

2

= Fx

2

M

2

x , x k er A .
2

f 0 R ( B Q ) , f k R ( B Q ) f k f 0 k . xk k e r A , f k = B xk .

fk - f

2
m

= B xk - B x

2
m

= B ( xk - x

m

)

2

M

2

xk - x

2
m

.

,

{ xk }

.

xk x

0

(k )

, x0 k e r A ,

f k = B xk , f 0 = B x0 . f 0 R ( B Q ) .
, R ( B Q ) = R ( B Q ) . (3) (12) .

122


:

1.

§ 10

(12) (3).

z H : z = x + y , x k e r A ,
y k er A . C : H H : Cz = Ax + B y .
, C : H H ) , ) : k e r C = {0} , ) «»:

R (C ) = H .
) A B . C z = 0 , A x + B y = 0 , z = x + y . x k e r A , y k er A . (10.45), -, , A x = 0 , ,

x = 0 , - B y = 0 y k er B . y k er A y k er B , (10.44) , y = 0 . , z = 0 . ) . ) (10.46). C : H H ), ) ), [43] , M > 0 ,

C - 1w M w

w H .

(10.48)

w H ­ . (10.46) u R ( A ) v R ( B Q

)

, w = u + v . z = C -1w .

z = C -1u + C -1v .
C
- B0 1 , : - A0 1 = C -1
R( A
-1

(10.49)

- : H H R ( A ) R ( B Q ) A0 1

)

,

- B0 1 = C

-1
R( B Q

)

.

(10.50)

(10.49), (10.50),
- - z = A0 1u + B0 1v .

123


:

1.

§ 10

(10.48)
- A0 1u M u , - B0 1v M v .

(10.51)



u0 R ( A) .







- u k R ( A ) , u k u 0 k . xk = A0 1u k . (10.51)



{ xk }

.

xk x

0


0

k . A , A xk A x
k . A xk = u
k

u k u 0 k , A x0 = u 0 . -

u 0 R ( A ) . A . F0 , k er A k e r A . y k er A F0 y = 0 . P B y = 0 , ,

By ker P = R ( A) = R ( A) .

(10.52)

, B y = B Q y R ( B Q ) . (10.52), (10.45), B y = 0 . , y k er B . , y k er A , (10.44), y = 0 .

ker F

k er A

= {0} .

f k er A ­ . P f = f . (10.46), x k e r A y k er A , f = A x + B y . f = P f = P A x + P B y = P B Q y = F0 y . , F0 : k e r A k e r A «». , F0 : k e r A k e r A
-1



, «». [43] F0 : k e r A k e r A . 124


:

1.

§ 10

(12) (3) . 10.1 . 10.1 1. 10.6. 10.2-10.4. 10.2. (8) (10.19) (10.20) , (3) (10.1) (10.2). 10.1 (3) (8) . (1) (3) . 10.2 . 10.3. (10.2) (10.1). 10.1 (3). F0 , k er A

k e r A .
10.3 . 1

( 1)
( 9)

( 4)

( 5) ( 3)
( 2)

( 8)

(

13

)

(

12

) (
10

( 6)

)

(

11)

( 7)

125


:

1.

§ 11

10.4. e

{ } {g }
n n

­ -

k er A k e r A , . B , en k er A g n k e r A

k er A : B en = g
n

en k e r A



Bx = 0 x k er A .

10.4 .

11. A ­ n r : r < n .

A1 1 A= A21
A1 1 , A2
2

A1 2 , A22

(11.1)

­ r n - r , .

A1 1 ­ ,

d e t A1 1 0 ,
A22

(11.2)

A22 = A21 A1-11 A12 ,

(11.3)

A1 1 A . A (11.1) A1 1 . A , H .

126


:

1.

§ 11

A : H H ­ . H

H = R A k e r A, H = R ( A ) k e r A ,

()

(R( A )


= ( ker A

)



)

, .

(

R ( A) = ker A

(



)

)

H ( )

H = H1 H 0 ,
,

H = F1 F0 ,

(11.4)

E - Q : H 1 R A ,

()

Q : H 0 ker A ,

(11.5) (11.6)

E - P : F1 R ( A ) ,

P : F0 k e r A



«». Q P ­ k er A k e r A . (11.4) ( , )

Q1 : H H 1 , P1 : H F1 ,
:

Q0 : H H 0 , P0 : H F0 ,

· x = x1 + x0 , x1 H 1 , x0 H 0 ,

Q1 x = x1 ,

Q0 x = x0 ,

· f = f1 + f 0 , f1 F1 , f 0 F0 ,

P1 f = f1 ,
127

P0 f = f 0 .


:

1.

§ 11

,





.



Q1 , Q0 = E - Q1 , P1 , P0 = E - P1 , ,


Q1 = E - Q0 : R A

( )

H1,

Q0 : k e r A H 0 ,

(11.7) (11.8)

P1 = E - P0 : R ( A ) F1 ,

P0 : k e r A F0 ,

«». (11.5)-(11.8) «», H 1 , H 0 , F1 , F0 . A : H H

H1 = R A

( )

= ( ker A

) , H 0 = ker A
,

,

F1 = R ( A ) = k e r A

(



)

F0 = k er A .

. H 1 , H 0 , F1 , F0 ­ .

Ai j = Pi A Q j : H j Fi ,
A

i, j = 0, 1 ,

(11.9)

A1 1 A= A01

A1 0 . A00

(11.10)

H 1 , H 0 , F1 , F0 ­ ,

A1 0 = 0, A01 = 0, A0 0 = 0 , ,

A A= 1 0

1

0 . 0

128


:

1.

§ 11

(11.10) A . H (11.4) , A . 11.1. H 1 , H 0 , F1 , F0 ­ . A1 1 = P A Q1 : H 1 F1 . 1 . , A1 1 :

k e r A11 = {0} .
A11 x1 = 0 , x1 H 1 .

(11.11)

P1 A Q1 x1 = 0 .

(11.12)

x1 H 1 , Q1 x1 = x1 , , (11.12)

P1 A x1 = 0 .









P1 : R ( A ) F1 A x1 R ( A ) A x1 = 0 .
(11.13)

x1 k e r A . A Q . (11.13)

A ( E - Q ) x1 = 0 .

(11.14)

E - Q ( k e r A

)



,

( E - Q ) x1 ( ker A) ( E - Q ) x1 = 0



.







(11.14)





. E - Q : H 1 R A

( )

-

. (11.11) .

( E - Q ) x1 = 0

x1 = 0 .

129


:

1.

§ 11

, A1 1 «»:

A11 H 1 = F1 . Q1 H 1 = H 1 .
A Q , E - Q : H 1 R A «».

(11.15)

Q1 : H H 1 , (11.4) (11.16)

( )

-

AH1 = A( E - Q ) H1 = AR A
«»,

( )

= R ( A) .

(11.17)

, , P : R ( A ) F1 1

P1R ( A ) = F1 .
(11.16)-(11.18) (11.15). , , A1
1

(11.18)



H 1 F1 . , [43], A1 1 : F1 H 1 . 11.1 . x0 k e r A ­ .
-1

H = H 1 H 0 , x0 = z1 + z 0 , z1 = Q1 x0 H 1 , z 0 = Q0 x0 H 0 .
A x0 = 0 :

A11 z1 + A10 z 0 = 0, A01 z1 + A0 0 z 0 = 0,
Ai
j

(11.19)

(11.9). A1 1 ­ -

, , «» (11.19)

- A01 A1-11 ,

(

A00 - A01 A1-11 A10 z 0 = 0 .
130

)

(11.20)


:

1.

§ 11

z 0 = Q0 x0 , x0 k e r A . , A00 - A01 A11 A10 H 0 ,
-1

{

z 0 = Q0 x0 : x0 k er A} H 0 .
-1

(11.20) A00 - A0 1 A1 1 A1 0 = 0 ,

A00 = A01 A1-11 A10 .
,

(11.21)

11.2. H 1 , H 0 , F1 , F0 ­ . (11.21). (11.21) (11.3) . , A (11.10), (11.9), , (11.2) (11.3). , A1 1 , , .

A11 = P1 A Q1 : H 1 F1 ,

(11.22)

(11.21), A : H H , H 1 ,

H 0 , F1 , F0 .
(11.19)

A x + B x = f . Bi j = Pi B Q j , i, j = 0, 1 , B : H H


B1 1 B= B0 1

B1 0 . B0 0

131


:

1.

§ 11

A + B .

C1 1 C10 A + B = , C 0 1 C 00
C
ij

=C

ij

( ) Ai j + Bi j ,
A1
1

j , j = 0, 1 .
A ,



­

C1 1 ( ) A1 1 + B1 1 B1 1 , A1-11 B11 0 < 1 .

C1-11 ( ) = ( A1 1 + B1


1

)

-1

C1-11 ( ) = ( A11 + B1
M
11

1

)

-1

M 11 ,

= A1-11 (1 -

0

)

-1

.

C1-11 ( ) = A1-11 - C1-11 ( ) B11 A1-11 ,


C1-11 ( ) - A1-11 M

11

B11 A1-11 .
-1

A + B

0 , ( A + B )

,

(. [25], . 59) :

( A + B ) -1 =

D1

1 1

D0

D10 , D00

(11.23)

132


:

1.

§ 11



D11 = D11 ( ) C1-11 +
+C1-11C10 C 0 0 - C 01C1-11C1 D10 = D1 1 ( ) -C1-11C10 C

(

0

)

-1

C 0 1C1-11 ,
0

(11.24)

(

00

- C 01C1-11C1
0

)

-1

,

(11.25) (11.26) (11.27)

D01 = D0 1 ( ) - C 00 - C 0 1C1-11C1

(

)

-1

C 01C1-11 ,

D00 = D00 ( ) C 00 - C 0 1C1-11C1

(

0

)

-1

.

, , (11.23)-(11.27), A + B , . 11.3. T : H H , S : H H , 0 : T M 1 , S


M 2 , 0 .

T + S

{(

T + S



)

-1

}

(

)

-1



, 0 :

(

T + S



)

-1

M , 0 ,

(11.28)

T . . . T + S

(



)

-1



{(

T + S



)

-1

}

, -

. T . , T :

k e r T = {0} .

(11.29)

133


:

1.

§ 11

, T x0 = 0 . , x0 = T + S

(

T + S



)

x0 = S x

0

, -

(



)

-1

S x0 , 0 . -

(11.28), x0 = 0 . (11.29) . , T

R (T ) = H .
T + S

(11.30)

, f H ­ . -

(



)

x = f . x = (T + S



{ x }



)

-1

f , 0 . (11.28)

, 0 .

{n }

­ -

: n 0 ( n ) .

{x }

n

-

. ,

{x }

n

x0 H .

( )

f R (T ) . (11.30) . (11.29) (11.30) [43] T : H H . . . T : H H ­ . 0 > 0 , 0 <

(

T + n S



n

)

n

x



n

= f , ( n ) , T x0 = f . -

{(

T + S



)

-1

}

0

.

,

0 > 0





0 T

-1

M 2 < 1 . , 0 < 0

{(

T + S



)

-1

}

T -1S



T
134

-1

S



0 T

-1

M 2 < 1.


:

1.

§ 11

E + T

-1

S



,

(


E + T -1S



)

-1



1 1 - 0 T
-1 -1

-1

M

.
2

(

T + S



)

-1



T

1 - 0 T

M

.
2

. 11.3 . 11.4. H 1 , H 0 , F1 , F0 ­ . D0

{

0

( )}
(11.31)

D00 ( ) N 00 ,


D1 B00 - A01 A1-11 B10 - B0 1 A1-11 A1 0 + A01 A1-11 B1 1 A1-11 A1 0 .
. .

(11.32)

{

D00 (

)}

. D1 .

D00 ( ) = D0 + D1 - 2 D2 (


(

)

)

-1

,

D0 = A0 0 - A01 A1-11 A10 ,
D2 ( ) = B01C1-11B10 - A01C1-11 B11 A1-11 B10 - B0 1C1-11B1 1 A1-11 A1 0 + + A0 1C1-11 B11 A1-11B1 1 A1-11 A1
0

(

)

-1

,

(11.33)



D2 ( ) M 2 .
135

(11.34)


:

1.

§ 11

A1 1 ­ A , (11.21). D0 = 0 D0
0

( )



D00 ( ) = ( D1 - D2 ( )
D1-1 N1 . . .

)

-1

.

(11.35)

(11.31) (11.34) > 0 -

D1

:

D1-1 N1 . 0 > 0 , 0 < 0 D0

{

0

( )}

.

(11.35).

(
D1-
1

D1 - D2 (

)

)

-1

= E - D1-1D2 (

(

)

)

-1

D1- 1 ,

M 2 0 < 1

(

D1 - D2 (

)

)

-1



D1-

1

1-

.
0

. 11.4 . .

{ {

11.1.

( A + B)

-1

} }

,

0

-

(11.32) ­ . . .

( A + B)

-1

, H 1 , H 0 , F1 , F0 ­ 136


:

1.

§ 11

. (11.32). 11.1 A1 1 .

( A + B)

-1

(11.23)-

(11.27).

{

( A + B)

-1



0



{

Di

j

(

} )}

, ,

0 i, j = 0, 1 . , 11.3, (11.32). . . H 1 ,

H 0 , F1 , F0 , (11.32), . 0 > 0 ,

{

0 = { : 0 <

( A + B)

-1

}

0

}

A + B ,

.
1

11.1 A1 . 1 > 0

-

1 A1-11B11 < 1 .
: 0 < 1 E + A11 B11 . C1 1 = A1 1 + B1 1 .
-1

(

E + A1-11B1

1

)

-1

<

1 1 - 1 A1-11B1
A1-11
1

,

(

A1 1 + B1

1

)

-1

<

1 - 1 A1-11B1

.
1

137


:

1.

§ 11

D1 - D2 ( ) , D1 D2 ( (11.32) (11.33), D2 (

)

)



(11.34). D1 . N1 > 0 , D1-1 N1 . 2 > 0 2 N1M 2 < 1 , 0 , 0 = m i n 1 , D1 - D2 (

{

2

}

, -

)

. ,

0 E - D1-1D2 ( )
D1 - D2 ( ) ,

(

E - D1-1D2 (

)

)

-1

<

1 , 1 - 0 N 1M 2

(

D1 - D2 (

)

)

-1

<

N

1 - 0 N 1M

1

.
2

(11.36)



C 00 - C 01C1-11C10 = ( D1 - D2 ( ) ) .
C
00

- C 01C1-11C10 , (11.36) ­ :

(

C00 - C

-1 C -1C 0 1 11 1 0

)

<

1 - 0 N1M
0

N1

-1

.
2

, D0

{

0

( )} ,

-

. (11.24)-(11.26) D1 1 (

{

)} , { D10 ( )} , { D01 ( )} ,

(11.23) . 11.1 .

{

0

,
-1

( A + B)

}

.

138


:

1.

§ 12

12. 12.1. . , A, B, f

f A, B, , A - A µ, B - B , - f f , f
(12.1)

µ 0, 0, 0 ­ ( ). , ( )

Ax = f ,

(12.2) , (12.3)

( A + B) x = f
A x = f

(
(12.2). x

A + B) x = . f

(12.4)

, (12.3)

(12.3). -

x x0 0 , x0 ­ (12.2): A x0 = f . , : 1) (12.4)

= ( , A, B ) = ( A + B x x f

)

-1

f

f 0 A, B, ;
2)

l i m x = x0 ;
0

(12.5)

3) (12.5).

139


:

1.

§ 12

H 1 , H 0 , F1 , F0 ­ . A, B, f

A1 1 A= A01

A1 0 B11 , B = A0 0 B0 1

B10 f1 , f = , B0 0 f0

f A, B, ­ A A = 11 A01 B11 A1 0 , B = A0 0 B0 1 1 B10 f f , = . B0 0 f0

k = k H 1 , H 0 , F1 , F0 , (12.1)

(

)

Aij - Ai j k µ ,

Bij - Bij k ,

- f k f , i, j = 0,1 fi i

.

(12.6)

12.2. . .

T1 = B00 - A01 A1-11B10 - B01 A1-11 A10 + A01 A1-11B11 A1-11 A10 .
12.1.

(12.7)

0,


µ 0, 0 .


(12.8)

{



(

A + B

)

-1

}

,
0



-

T1 : H 1 F1 . 12.1 12.4.

140


:

1.

§ 12

12.1 ,

A , µ , , µ / 0

0 . B ­ . µ
,

µ / 0 0 , .
. , -

A .

Aij - Ai j = µ Aij + o ( µ ) ,

i, j = 0, 1 ,

(12.9)

Aij : H j Fi , i, j = 0, 1 ­ .

T = A00 - A01 A1-11 A10 - A01 A1-11 A10 + A01 A1-11 A11 A1-11 A10 .
, T H 1 F1 . 12.2. = µ 0 , 0 .

( A + B

)

-1

D







D : H 1 F1
T1 : H 1 F1 T = 0 . 12.2 12.4. , 12.2, 12.1. , 12.1 T1 : H 1 F1 , 12.2 T = 0 ; 141


:

1.

§ 12

12.1

A + B

{

(

)

-1

}

, 0 , -

12.2 . , µ 12.1 : µ = o ( ) , 12.2 µ ,

: µ = .
. µ

µ , ­
0, 0 , 0 > 0 ­ .

T2 = ( A00 - A0

0

)

- A01 A1-11 ( A10 - A10 ) - ( A01 - A01 ) A1-11 A10 + + A01 A1-11 ( A11 - A11 ) A1-11 A10 .
(12.10)
-1 1

(12.6) A01 , A1 1 A1 m2 > 0 , ,



T2 m2 µ .
12.3.

(12.11)

0,

µ , 0,

(12.12)

0, 0 ­ , 0 ­ .

A + B

{

(

)

-1

}

,

0

-

T1 : H 1 F1 .

142


:

1.

§ 12

12.3 12.4.

A + B

(

A+

( 0, -1 B)

0 , 0 > 0 ­ ,


(

A + B

)

-1

D ( = 11 D0 1 (

) )

D10 ( ) , D00 ( )

(12.13)

Dij ( ) , i, j = 0, 1 (12.42) ­
(12.45). (12.4)

x ,1 D1 1 ( ) 1 + D1 0 ( ) 0 f f = x = . f f x ,0 D0 1 ( ) 1 + D0 0 ( ) 0
, f R ( A )
-1 -1 -1 -1 -1 x0,1 A1 1 f1 - A11 A1 0T11 A1 0 A1 1 B11 - B01 A1 1 x0 = = x0 , 0 T1-1 A1 0 A1-11B11 - B01 A1-11 f1 1

(12.14)

(

(

)

)

f1 .

(12.15)

(12.2). 12.4. f R ( A ) , T1 : H 1 F1

0,

µ , 0,

0,

0, 0 ­ , 0 ­ .

x - x0 M 1 + + ,
M 1 > 0 ­ . 143


:

1.

§ 12

12.3 12.4. 12.3. . :

Cij = C

ij

( ) Aij + Bij , Cij = Cij ( ) Aij + Bij , i, j = 0,1

.

C
ij

Cij . -

0 = : 0 < µ 0 µ µ

{

0

}

, 0 > 0 ­

, , 0

0 0 ,

µ 0 > 0 0 > 0 ­ .
A , B , A , B «» (12.1) (12.6)
, µ m0 > 0 ,

C

ij

m0 ,

C i j m0 ,

i, j = 0, 1 .

(12.16)

12.1. (12.6).

Cij - C

ij

k ( µ + ),

i, j = 0, 1 .

(12.17)

. 12.2.

0 A1-11B11 µ q1 < 1 .
0 = : 0 <

(12.18)

{

0

}

C1 1

C1-11 ( )

A1-11 1 - q1

,

0 ,
B1
1

(12.19)

A1-11

-

C1-11

( )

A1-11

2

1 - q1
144

,

0 .

(12.20)


:

1.

§ 12

. 0

C1-11 ( ) = A1-11 - E + A1-11 B1


( (

1

) )

-1

A1-11B1 1 A1-11 . A1-11B1 1 A1-11

A1-11 - C1-11 ( ) = E + A1-11 B1


-1

1

(12.21)

(

E + A1-11B1

1

)

-1



1 . 1 - q1

(12.22)

C1 1 ( ) = A1

1

(

E + A1-11B1

1

)

-

C1 1 ( ) , (12.22) ­ (12.19). (12.21) (12.22), (12.20). . 12.3.

A1-11 µ q 2 < 1 .

(12.23)

A1 1 : H 1 F1


A1-11

A1-11 1- q
2

.

(12.24)

.

A1 1 = A11 E + A1-11 ( A1 1 - A1


(

1

))

(12.25)

A1-11 ( A11 - A1

1

)

A1-11 A11 - A1 1 A1-11 µ q 2 ,

(12.26)

145


:

1.
-1 1

§ 12

(12.23), E + A1 (12.26),

(

A11 - A11 ) ,

(


E + A1-11 ( A1 1 - A1

1

))

-1



1 . 1 - q2
,1 1

(12.27)

, (12.25), Aµ

-

A1-11 = E + A1-11 ( A1 1 - A1

(

1

))

-1

A1-11 .

(12.28)

(12.28) (12.27), (12.24). . 12.4.

A1-11 µ +

(

0

(

B1 1 +
0

0 = : 0 <

{

) ) q3 < 1 . }

(12.29)

C1 1 = C11 ( , µ , ) A11 + B11 : H 1 F1


C1-11

A1-11

1- q

,

(12.30)

3

C1-11

-

A1-11



A1-11

2

1- q

(

µ + ( B11 +

))

.

(12.31)

3

.

C1 1 = A11 ( E + T1 T1 1 = A1
-1 1

1

)

,

(12.32)

(

A1 1 - A1 1 ) + B1 1 ,

146


:

1.

§ 12

T1 1 A1-11 µ +

(

(

B11 +

))

q

3

E + T1 1

(


E + T1

1

)

-1



1 . 1 - q3

(12.33)

, (12.32), C1 1 C1-11 = ( E + T1

1

)

-1

A1-11 .

(12.34)

(12.34) (12.33), (12.30).

C1-11 - A1-11 =

(

(

E + T1

1

)

-1

-E

)

-1

A1-11 ,

(12.31). .

C

00

- C 01C1-11C10
(12.35)

C00 - C 01C1-11C10 = T0 + T1 + T2 + T3 , T1 T2 (12.7) (12.10),
T0 = A00 - A0 1 A1-11 A1 0 .

T2 T3 T2 m2 µ ,
m2 = k 1 + m0 A1-11

(

)

2

,

(12.36) (12.37)

2 T3 m + + µ 2

(

2

)

,

m > 0 ­ . 2 147


:

1.

§ 12

m1 =

(1

- q1 ) (1 - q

k A1-11

2

3

)

, q1 q3

12.2 12.4. 12.5. (12.29).

C1-11 - C1-11 m1 ( µ + ) .

(12.38)

. (12.29) (12.18). C1
-1 1



C1-11 . C1-11 - C1-11 = C1-11 ( C11 - C11 ) C1-11
(12.19), (12.30) (12.17), (12.38). . 12.6.

T1-1T2 q 4 < 1 .


(12.39)

(

C00 - C

-1 C -1C 01 11 10

)



(1 - q

T1-

1

4

)

.

(12.40)

.

C 00 - C 01C1-11C10 = T1 E + T1-1T2 + O
(12.39) E + T1-1T2 + O

(

( 2 ))

.

(12.41)

( 2)

.

(

E + T1-1T2 + O
148

( 2 ))

-1



1 . 1 - q4


:

1.

§ 12

(12.41) (12.40). .

D00 = C 0 0 - C 01C1-11C1

(

0

)

-1

,

(12.42) (12.43) (12.44) (12.45)

D01 = - D00C01C1-11 , D10 = -C1-11C10 D00 , D11 = C1-11 + C1-11C10 D00C 01C1-11 .
12.7.

0,

µ 0, 0 .

(12.46)

D0
T1 . . .

{

0

}



{ D00 }

-

. T1 . 11.2 T0 = 0 . (12.35)

C 00 - C 01C1-11C10 = T1 +
(12.36) (12.37),

(

-1

(

T2 + T3

))

.



-1

(

µ µ2 T2 + T3 ) m2 + O + + .

, (12.46),



-1

(

T2 + T3 ) 0 .
149

(12.47)


:

1.

§ 12



{ D00 }

, ,

(12.47) 11.3, T1 . . . T1 . -

D0


{

0

}

.

11.2 T0 = 0 . (12.35)

C 00 - C 01C1-11C1 0 = T1 E + -1T1-1 (T2 + T3

(

))

.

(12.48)

q5 < 1 - . > 0 , µ 0

0 (12.46)
µ µ2 -1T1-1 (T2 + T3 ) T1-1 m2 + O + + E + -1T1-1 (T2 + T3 ) q5 .



(

E + - 1T1- 1 (T2 + T3

))

-1



1 . 1 - q5

(12.48)

C 00 - C 01C1-11C10

(
{
D0
0

C 00 - C 01C1-11C1

0

)

-1



m3



, m3 =

T1 1- q
5

.



}

. .

.

150


:

1.

§ 12

12.8. H 1 , H 0 , F1 , F0 ­ .
( D00 D0
0 0

)



( = µ 0 , 0 D00) : H 1 F1 0













T1 : H 1 F1 T = 0 .
. .
( 0)





D00 D0

( 0)
0

= µ 0, 0 D00 : H 1 F1 ­

. T1 T = 0 .

(12.9) T2 T2 = µT + o ( µ ) ,


T = A00 - A01 A1-11 A10 - A01 A1-11 A10 + A01 A1-11 A11 A1-11 A10 .
, 11.2, (12.35)

(12.49)

C00 - C 01C1-11C10 = T1 + µT + o ( µ ) + T3
, ,

(12.50)

D00 = ( C 00 - C 01C1-11C1


0

)

-1

µ µ = T1 + T + o

1 + T . 3

-1

{ D00 }

= µ 0 , 0 .

, = µ , , = 1 ,

D00 (T1 + T

)

-1

= µ 0 , 0 .

(12.51)

­ , (12.51), , T = 0 151


:

1.

§ 12

(0 D0 0) ­ T1 : H 1 F1 . -

. . T1 : H 1 F1 T = 0 .
( D00 D0
0 0

)



= µ 0, 0



( D00 ) : H 1 F1 . 0

, (12.50) ,

C00 - C 01C1-11C10 = T1 + o ( µ ) + T3 .


D00 = ( C 00 - C 01C1-11C1


0

)

-1

µ = T1 + o

1 + T . 3

-1

(12.52)

µ 1 o + T3 0



= µ 0, 0 ,

, (12.52) = µ 0 , 0 ,

D00 T1-1 = µ 0 , 0 .
(0 , = µ 0 , 0 D00 D00) (0 D00) = T1- 1 ­ . .

. 12.9. (12.12). -

D0

{

0

}



T1 : H 1 F1 .

152


:

1.

§ 12

. .

{ D00 }

-

. T1 . 11.2 T0 = 0 . (12.35)

C 00 - C 01C1-11C10 = T1 +


(

-1

(

T2 + T3

))

.



-1

(

1 1 2 T2 + T3 ) m2 µ + m + + µ 2





(

2

)

.

, µ ,



-1

(

T2 + T3 ) m2 + m + 1 + 2

((

2

))

.

0 0


D0
.

-1

(

T2 + T3 ) 0 .

(12.53)

{

0

}

, (12.53)

11.3 T1 . . T1 . -

D0


{

0

}

.

11.2 T0 = 0 . (12.35) -

C 00 - C 01C1-11C1 0 = T1 E + -1T1-1 (T2 + T3


(

))

.

(12.54)

-1T1-1 (T2 + T3 )


1



T1-1 m2 µ +

1



2 T1-1 m + + µ 2

(

2

)

.

(12.55)

153


:

1.

§ 12

, µ ,

-1T1-1 (T2 + T3 ) T1-1 m2 + T1-1 m + (1 + 2

(

2

))

.

q6 < 1 - . T1-1 ,

m2 m 0 > 0 , 0 > 0 1 : 0 < 1 0 , 2

, ,
0 0 , 0 0 , 0 < 1 ,


T1-1 m2 + T1-1 m + (1 + 2
(12.55)

(

2

))

q6 .

-1T1-1 (T2 + T3 ) q6 < 1 .
E + T1
-1 -1

(

T2 + T3 ) -

(

E + - 1T1- 1 (T2 + T3

))

-1



1 1- q

6

, (12.13) ­

C 00 - C 01C1-11C10

(
{ D00 }

C 0 0 - C 01C1-11C1

0

)

-1



(1 - q

T1-

1

6

)

.

. .

.

154


:

1.

§ 12

12.10. f R ( A ) (12.29).

- C C -1 m ( µ + + + f 0 0 1 1 1 f1 5

)

f,

(12.56)

1 -1 µ C 01C11 f1 - f 0 - B01 A1-11 f1 k k 0 A1-11 +

(

)

+

A1-11

m B A -1 1 + 0 11 1 1 1 - q3

+ k 0 f ,

(12.57)



m5 = ma x k 0 , k 0 A1-11 , k1 ,

{

}

m A -1 k 0 = k 1 + 0 11 1 - q3

,

B A -1 k1 = k A1-11 B01 + 11 11 1 - q3 f R( A

.

.

)



f 0 = A01 A1-11 f1 .
- C C -1 = ( - f f 0 01 11 f1 f0
+C
01 0

)+(

C01 - C01 ) A1-11 f1 - B01 A1-11 f1 +

(

A1-11 - C1-11 f1 + C01C1-11 ( f1 - 1 ) . f

)

(12.58)

, (12.6) , (12.16), (12.17) (12.31),

- C C -1 k + ( µ + f 0 01 11 f1

)

A1-11 + B0

1

A1-11 +

+

m0 A1-11 1- q
3

2

(

µ + ( B11 +

))

+

m0 A1-11
1- q
3

f=

= m5 ( µ + + +
155

)

f.


:

1.

§ 12

(12.56) . (12.57). (12.58)

(

f f C 0 1C1-11 1 - 0 - B0 1 A1-11 f1 = ( f 0 - 0 ) + ( C 01 - C f +C
01

)

01

)

A1-11 f1 +

(

C1-11 - A1-11 f1 + C 0 1C1-11 ( 1 - f1 ) . f

)

, (12.16), (12.17) (12.31), (12.57). . 12.11. (12.29).

(

C 0 0 - C 01C1-11C1

0

)-(

C 00 - C 01C1-11C1

0

)



m A -1 k 1 + 0 11 1 - q1



m A -1 1 + 0 1 1 1 - q3

( µ + ) .

(12.59)

.

(

C 00 - C 01C1-11C1

0

)-(

C 00 - C 01C1-11C1

0

)=(

C 00 - C

00

)

-
0

- ( C 01 - C 01 ) C1-11C10 - C

01

(

C1-11 - C1-11 C10 - C 01C1-11 ( C10 - C1

)

)

12.2 12.4, (12.59). .

T2 = - B01 A1-11B10 - A01 A1-11B11 A1-11B10 - - B01 A1-11B11 A1-11 A10 - A01 A1-11B11 A1-11B11 A1-11 A10 , S S
01 00

( ) = -1T1-1 - T1-1T2T1-

1

,

( ) = - -1T1-1 A01 A1-11 + (T1-1 A01 A1-11B11 A1-11 +
+ T1-1T2T1-1 A01 A1-11 - T1-1B01 A1-11 ,
156

)


:

1.

§ 12

S1 0 ( ) = - -1 A1-11 A1 0T1-1 + A1-11B11 A1-11 A10T1-1 + + A1-11 A1 0T1- 1T2T1- 1 - A1-11B1 0T1- 1 . S1 1 ( ) = -1 A1-11 A10T1-1 A01 A1-11 + A1-11 + A1-11 A10T1-1B01 A1-11 -

(

)

(12.60)

(

- A1-11 A10T1-1 A01 A1-11B11 A1-11 - A1-11 A10T1-1T2T1-1 A01 A1-11 + + A1-11B10T1-1 A01 A1-11 - A1-11B11 A1-11 A10T1-1 A01 A1-11 .

)

(12.61)

12.12. (12.29) (12.18).

Dij = S ij + S i(j1) , Dij = S ij + S i(j1) , i, j = 0,1 ,

(12.62)

Dij Dij (11.24)-(11.27) (12.42)-(12.45)
1 1 , S i(j ) = S i(j ) (

)

1 1 S i(j ) = S i(j ) ( , µ ,

)

-



S i(j1) m6 ,

S i(j1) m

6

( +µ + )

,

(12.63)

m6 > 0 m7 > 0 ­ . 12.12

C 00 - C 01C1-11C10 , C 00 - C 01C1-11C10
, µ , . 12.13. f R ( A ) , T1

0,

µ , 0,

0,

(12.64)

157


:

1.

§ 12

0, 0 ­ , 0 ­ . x (12.4)


x (12.3)
+X
,0

x

,1

=x

0 ,1

+X

,1

,x

,0

=x

0,0

, ,

(12.65) (12.66)

x ,1 = x
x
0 ,1

0 ,1

+X

,1

, x , 0 = x

0,0

+X

,0

x

0,0

(12.15),

X
X

,1

m7 f ,

X

,0

m7 f ,

(12.67)

,1

m8 + + f , X

,0

m8 + + f . (12.68)

. (12.65) (12.67). (12.62)

S x = S

11 01

( ) f1 + S10 ( ) f 0 S1(1) ( ) f1 + S1(1) ( ) f 0 1 0 + (1) (1) ( ) f1 + S 0 0 ( ) f 0 S 0 1 ( ) f1 + S 0 0 ( ) f 0
-1

.

f R ( A ) , f 0 = A0 1 A1 1 f1 . (12.60), (12.61)

S1 1 ( ) f1 + S

10

( ) f 0 x0,1 , S 01 ( ) f1 + S 00 ( ) f 0 x

0,0

,

(12.69)

(12.63) ­ (12.67). (12.66) (12.68). (12.62)

S = x S

11 01

( ) f 1 + S1 0 ( ) f 0 + ( ) f1 + S 0 0 ( ) f 0
-f f0 -f f0
158
0

(12.70)

S ( )( - f ) + S ( )( f1 1 11 10 + - f ) + S ( )( S ( )( f 1 1 00 01

0

) + S1(1) ( 1 ) + + S 0(11)

) 1 + S1(1) ( ) 0 f 0 f
f ( 0 f ( ) 1 + S 01) ( ) 0

.


:

1.

§ 12

(12.69). (12.68) (12.63), (12.64) (12.6). . 12.4. 12.1 ­ 12.4. 12.1. . -

> 0 , µ 0 0 A + B
.

{

(

)

-1

}





T1 : H 1 F1 .
H 1 , H 0 , F1 , F0 -

A + B A + B
C1 A + B = C1

1 0

C10 , C 00

C A + B = 1 C1

1 0

C10 . C 00

(

A + B

)

-1

(12.13)

(12.42)-(12.45). A + B

{

(

)

-1

}

-

(12.8) , ,

{ D00 }

. 12.7 -

T1 . . . T1 : H 1 F1 .

{

(

A + B

)

-1

}

> 0 , µ 0 0 , (12.46). T1 : H 1 H 1 11.4

D0

{

0

}

. ,

(12.43)-(12.45), 159

{ D01}

,


:

1.

§ 12

{ D10 } { D11}

. , (12.13) -

A + B
12.1 .

{

(

)

-1

}

. .

12.2. .

A + B D = µ 0, 0 D : H 1 F1 ­ . T1 T = 0 .

(

)

( A + B ) D

=µ 0,

0 , , D00 D0 D0 : H 1 F1 . 12.8 T1 : H 1 F1

T = 0 . .
. T1 : H 1 F1 T = 0 .

( A + B

)

-1

D = µ 0, 0

D : H 1 F1 .
(0 12.8 D00 D00)

= µ 0, 0 . , (12.43)-(12.45),
( D01 D00 ) , 1
0 D10 D1(0 ) ,

0 D11 D1(1 ) = µ 0 , 0 ,

( ( ( ( D00 ) = - D00 ) A0 1 A1-11 , D00 ) = - A1-11 A10 D00 ) , 1 0 1 0
0 ( D1(1 ) = A1-11 + A1-11 A1 0 D00 ) A01 A1-11 , 0

160


:

1.

§ 12

. , (12.13)

( A + B

)

-1

D

0 D1(1 )

( D00 ) 1

0 D1(0 ) ( D00 ) 0

0 Di(j ) , i, j = 0, 1 ­ -

D . . 12.2 . 12.3 12.1. 12.4 12.9, (12.46) (12.12). 12.4. (12.69), (12.70) (12.68)

x ,1 - x x ,0 - x

0,1

X

,1

m8 + + f , m8 + + f .

0,0

X

,0

12.4 .

161


:

1. §§ 2 2 11

. , (12.13)

( A + B

)

-1

D

0 D1(1 )

( D00 ) 1

0 D1(0 ) ( D00 ) 0

0 Di(j ) , i, j = 0, 1 ­ -

D . . 12.2 . 12.3 12.1. 12.4 12.9, (12.46) (12.12). 12.4. (12.69), (12.70) (12.68)

x ,1 - x x ,0 - x

0,1

X

,1

m8 + + f , m8 + + f .

0,0

X

,0

12.4 .

161


:

2.

§1

2
1. L - 1.1. (). A : H F ­ ,

H F ­



H





F

, -

( , ). Ax = f , (1.1) f F ­ , x H ­ . K H ­ . (1.1) K [36, 37] x0 K ,

A x0 - f = i n f

{

Ax - f : u K

}

K . ... K H . (1.1) . (1.1) , , , . K = H , (1.1) . (1.1) (1.1). , (1.1) , , . 1.2. L -. A : H F L : H G ­ , f F g G ­ , H , F G ­ . ­ L - L -. , , ,

162


:

2.

§1

, . : · ( S ; f , g ) (. [65, 66]): x0 = x0 ( f , g )

Lx = g
(1.1). · (V ; f , g ) [59, 60]:

(1.2)





( x)

Ax - f

2

+ Lx - g

2

mi n , x H ,

> 0 , 0 . 1.3. . [58, 59], , A : H F L : H G , > 0 , x H

Ax + L x x , x H .
2 2 2

(1.3)

H , F G , , A L

k e r A k e r L = {0} .
A L , A L , , l
163
2

(1.4)

H , F G (1.4). , (1.4) .
2

(

H = F =G=l

)




:

2.

§1

A = d i a g A1 , . . . , An , . . . ,... L = d i a g L1 , . . . , Ln , . . . ,

(

)

(

)



1 1 1 + n An = , 1 1 + 1 n
k e r An = en : ,

1 1 Ln = , 1 1

n = 1, 2, . . . .



{

}

k e r Ln = g n : ,

{

}

( 1 .5 )



1 en =

1 + n, -1

gn = -

1 . 1

(1.6)

(1.5) , k e r An k er Ln = {0} , (1.4). A L . .
A A = d i a g A1 A1 , . . . , An An , . . . ,

(

)

L L = d i a g L L1 , . . . , L Ln , . . . , 1 n An An = 1 2 1 + n , 2 1 1 2 1 + 2 1 + n n 2

(

)

n = 1, 2, . . . ,

2 2 L Ln = , n 2 2

n = 1, 2, . . . ,

164


:

2.

§1

An An L Ln ( n

An Ln )
1 ( An An ) = 0, 2 ( An An ) = 2 + 2 1 + , n



1

2

n = 1, 2, . . . ,

1 ( L Ln ) = 0, 2 ( L Ln ) = 4, n n

n = 1, 2, . . . .

,
An An L Ln . , n

A A L L . A L . , A L (1.4) . , , A L . ,

xn = 1 , . . . , n-1 , g n , n+1 , . . . ,
k = ( 0, 0 ) , k = 1, 2, . . . , g
2
n

(

)

­ (1.6).

A xn

+ L xn

2

= An g

2 n

+ Ln g

2 n

=

2 , n2

n = 1, 2, . . . .

, > 0 , (1.3) x H . 1.4. L -. , , A A L . A A+ , k er A A ­ Q .

165


:

2.

§1

: 1.1. ( S ; f , g ) f F g G ; ( S ; f , g )

x0 = A+ f + ( L Q


)

+

(

g - L A+ f .

)

(1.7)

T0 = E - ( L Q

)

+

L,

T = - A A

(

)

+

L L ,

(1.8)

0 = (T ) ­ T .
1.2. (V ; f , g

)

f F ,

g G > 0 , 0 ,
; (V ; f , g )

x = x ( f , g ) A A + L L
: 0 < <
-1 0

(

)(
-1

A f + L g ,

)

( 1 .9 )



x - x0 m ,
x0 ­ ( S ; f , g ) , m > 0 ­ , . 1.1 ( S ; f , g ) . , , , .

166


:

2.

§1

1.3. ( S ; f , g ) f F g G A L Q ; ( S ; f , g )

x0 = A+ f + ( L Q

)

+

(

g - L A+ f + Q Q1 y ,

)

Q1 ­ L Q , y H ­ . 1.1 ­ 1.3 . 1.6. . . [2], . . [56, 57], . . [110 - 112] . 1.5. . , 1.1 ­ 1.3. 1.1. A : H F . A L : H G

L:ker A R ( LQ ) .

(1.10)

. . A L . (1.10). A 1 > 0 ,

Ax 1 x



x ker A .

x k er A , Q x = x , , (1.3)

L x = L Qx x , x k er A .
2 2 2

(1.10). .

167


:

2.

§1

. (1.10) . A : H F L : H G . . (1.3), xn H ,

xn = 1 , n , A xn

2

+ L xn

2

0



n.

(1.11)

xn = xn ,1 + xn ,0 , xn ,1 k e r A, xn ,0 k e r A ,
(1.11) A xn = A xn
,1

(1.12)

0 n . -

A

xn
xn
,0

,1

0



n.

(1.13)

1 n . (1.11) (1.12), = L Q xn 0

L xn

,0

,0



n.

(1.10)

xn

,0

0 n . (1.13) xn 0 n ,



xn . A L . . 1.1 . 1.1. . L Q : H G .

168


:

2.

§1

1.2. A : H F (1.10) . L Q : H G . , H = k er A k e r A ,

L Q ( H ) = L Q ( ker A) = R ( L Q ) .
(1.10) R ( L Q ) = R ( L Q ) . L Q ( H ) = L Q ( H ) , . 1.1. A L A . , L = E , (1.3) = 1 , A , , . 1.2. A : H F . A L : H G

(

LQ

)

+

LQ = Q .

(1.14)

. . A L . (1.14). , , .

(

LQ

)

+

L Q Q , (1.14) ker ( LQ



)

+

L Q = k er Q .
+

k e r Q k er ( L Q k e r ( L Q

)

L Q .

)

+

L Q k er Q .

169


:

2.

§1



x k er ( L Q

)

+

L Q . x k er L Q , ,

L Q x = 0 . x = x0 + x1 , x0 k er A = k e r Q , x1 k er A = k er Q .

A x0 = 0 , Q x = Q x0 = x0

L x0 = L Q x0 = L Q x = 0 .
A L

0 = A x0


2

+ L x0

2

x0 .

2

x0 = 0 . ,

x = x1 k e r Q . .
. (1.14). A L .

k e r A k e r L = {0} .
(1.14) x = Q x = ( L Q

(1.15)

x k e r A k e r L . Q x = x L Q x = L x = 0 . -

)

+

L Q x = 0 . (1.15) -

. > 0 , (1.3). . (1.3).

xn H , xn = 1 , n , (1.11).
xn (1.12). A xn = A xn
,1

0, 0,

n A xn

,1

170


:

2.

§1

n , , xn

,0

1 , n . , -

L xn 0 , n L

L xn

,0

= L xn - xn

(

,1

)

L xn + L xn

,1

0 , n . (1.16)

xn ,0 k e r A , Q xn ,0 = xn ,0 . (1.14)

xn ,0 = Q xn ,0 = ( L Q

)

+

L Q xn ,0 . , -

(1.16) 1.1

xn

,0

( LQ

)

+

L Q xn

,0

= ( LQ

)

+

L xn
,0

,0

0,

n.

xn . 1.2 .

1 , n .

1.3. A : H F . A L

Q Q1 = 0 ,


Q1 = E - ( L Q

)

+

LQ .

1.3. A : H F L : H G A A + L L > 0 ­

> 0.
. . A L .

A A + L L > 0 . , 0 < 1 ,
171


:

2.
2

§1

Ax + Lx x ,
2 2

> 1 ,

Ax + Lx x .
2 2 2

= m i n { , 1} ,

A x + L x x .
2 2 2

A A + L L > 0 . A A + L L .

(

)

-1

> 0 .

(

A A + L L

)

-1

-

> 0 . xn H , xn = 1 , n ,

(


A A + L L

)

-1

xn



n.

(1.17)

yn = A A + L L

(

)

-1

xn ,

n .

(1.18)



Ay


2
n

+ Ly

2
n

y

2
n

,

n ,

((
(

A A + L L yn , y

)

n

)

y

2
n

,

n .

, (1.18),

x , A A + L L n

)

-1

xn

(

A A + L L

)

-1

2

xn .

172


:

2.

§1



(

A A + L L

)

-1

xn :

A A + L L xn , A A + L L

( (

) )

-1 -1

xn xn

-1 A A + L L xn , 0 < < 1, . -1 A A + L L x , 1 . n

(

)

(

)

(1.19)

n

A A + L L xn , A A + L L

( (

) )

-1 -1

xn xn xn

( (

A A + L L A A+L L


) )

-1 -1

xn xn

=1,

(1.19) n , (1.17), 1 . A A + L L

(

)

-1

. .

. A A + L L > 0 . A L . . A L . xn H ,

xn = 1 ,

n , A xn
2

+ L xn

2

0



n.

A xn 0 , L xn 0 n , ,
173


:

2.

§1

((
(

A A + L L xn , xn 0

)

)



n.

A A + L L 0 > 0 ,

A A + L L xn

)

0



n .

, , xn xn , y
nk

k

0 k ,

yn = A A + L L xn .
k k

(

)

A A + L L

(

)

-1



xn = A A + L L
k

(

)

-1

y

nk

0



n,

xn . . 1.3 .

X X 0 = T0

( ( A A) (
-1 + m -1

= Q L L Q

)

+

,
-1

E - L L X

)

,

X m = TX

,

m 1,

T0 T (1.8). 1.4. A : H F A L : H G .

(

A A + L L

)

-1

=

m = -1





mXm ,

(1.20)

174


:

2.

§1

0 < < (1.8).

-1 0

, 0 = (T

)

­

. (1.20) 0 < <

-1 0



Xm = T m X0,

m 1.

(1.20)

A A T0 = A A ,

A A A A

(
)

)

+

= E -Q ,

A A T + L L = Q L L ,

Q L L ( L Q

)

+

= Q L .

(
:

A A T + L L T0 = 0 . = A A L QQ L
-1

A A X A A X 0 + L L X

-1

(

)

+

=0

-1

= A A T X

+ A A T0 A A + A A A A

(

)

+

+ L L X

-1

=

= A A T + L L X = Q L L Q L L Q

(

)

-1

(

)

+

=

(

)

+

+ E - Q = ( LQ

)

+

LQ + E - ( LQ

)

+

LQ = E

A A X 1 + L L X 0 = A A T X 0 + L L X 0 = = A A T + L L T0 A A
m 2

(

)(

)(
+

E - L L X

-1

)

=0,

A A X m + L L X

m -1

= - A A T + L L T0 A A
175

(

)(

)

+

L L X

m-2

=0.


:

2.

§1



(
= A A X
-1

A A + L L +

)



m= -1

mXm =

m = -1







m

(

A A X m + L L X

m -1

)

=E.

1.4 . 1.5. A : H F A L : H G .

> 0 (V ; f , g )
(1.9). . :


( x)

-





( x) = (

A x, A x ) + ( L x, L x ) -

- 2 ( A x, f ) - 2 ( L x, g ) + ( f , f ) + ( g , g ) .
h :


( x)

, -





(

x + h) -



( x)

= 2 ( A x, A h ) + 2 ( L x, L h ) -

- 2 ( f , A h ) - 2 ( g , L h ) + ( A h, A h ) + ( L h, L h ) = = 2 A A + L L x - A f + L g , h + + ( A h, A h ) + ( L h, L h ) .


((

)(

))

(

A A + L L x - A f + L g = 0 ,

)(

)

176


:

2.

§1



(

A A + L L x = A f + L g .

)

(1.21)

1.3 A A + L L > 0 . (1.21) f F g G > 0 (1.9). 1.5 . 1.4 1.5 (V ; f , g ) . 1.6. A : H F A L : H G .

f F g G (V ; f , g )


x ( f , g ) = E - ( L Q


(

)

+

L A+ f + ( L Q

)

)

+

g+

m =1





mT

m -1

x ,

+ x = -T0 A A L L T0 A+ f - E - L ( L Q

(

)

(

)

+

)

g .

:

A1 : H1 F1 L1 : H1 G1 ­ , H1 , F1 , G1 ­ .
1.7. A1 A1 L1 .

x f1 , 0 x0 f1 , 0
x0 f1 , 0 ­ ,

(

)

(

)



0,

(1.22)

(

)

x f1 , 0 = A1 A1 + L1 L1

(

)(

)

-1

A1 f1 ,

(1.23)

177


:

2.

§1

f1 F1 ­ , A1 . . . (1.22) f1 F1 . A1 . A1 , ,

k e r A1 = {0} .
(1.23) > 0

(1.24)

(


A1 A1 + L L1 x f1 , 0 A1 f1 . 1

)(
11

)


A1 f1





A1 A1 x1

(

f1 , 0 =

)

,

A1

( A x ( f , 0) - f )
1 1





0 ,

= 0 . (1.24)

A1 x1 f1 , 0 - f1 = 0 . ,

(

)

f1 = A1 x1 f1 , 0 R A1

(

)

()

, , R A1 = H1 . A1 . . . A1 . (1.22) f1 F1 . A1 A1 . A1 A1

()

-1

(

)

-1

A1 A1

(

)

-1 L1 L1

.

(

A1 A1

)

-1 L1 L1

m.

178


:


2.

§1

m q < 1 A1 A1 + L1 L1

(
( )

A1 A1 + L1 L1

)

-1

A1 f1 - A1-1 f1

m
1- q

A1-1 f1 . f1 F1 ,

(1.22)

x0 f1 , 0 = A1-1 f1 . .
1.7 . 1.8. A L .

x ( f , 0 ) A A + L L

(

)

-1

A f x0 ( f , 0

)

0 ,

x0 f1 , 0 ­ , f F ­ , , A L Q . .

(

)

H1 = k e r A, F1 = k e r A , A1 = A

H1

, L1 = L

H1

,

f = f 0 + f1 , f 0 k er A , f1 k er A .
x ( f , 0 ) x f1 , 0 = A1 A1 + L1 L1

(

)(

)

-1

A1 f1 .

A1 : H1 F1 . , 1.7. A1 . R ( A ) = R A1 A : H F . x = x ( f , 0 ) ­ (),
179

()

-


:

2.

§1

(

A A + L L x = A f .

)

(1.25)

A

x = A+ f + Q y0 + y ,

(1.26)

y0 ­ , y

y 0 : y y

0

(

0 ) . (1.26) (1.25), -

:

A A y + L L A+ f + Q y0 + L L y = 0 .
0 :

(

)

A A y + L L A+ f + Q y0 = 0 .
Q A = 0 , , Q ,

(

)

Q L L Q y0 = -Q L L A+ f .
f F .

Q L L Q = ( L Q ) L Q ,


L Q . 1.8 . 1.9. A L .

x ( 0, g ) A A + L L
x0 ( 0, g

(

)

-1

L g x0 ( 0, g

)

0 ,

)

­ , g G ­ ,

L Q A .

180


:

2.

§1

. . (V ; f , g ) : x x ( 0, g ) x 0 , x H ­ . L Q . x x :

x = x ,0 + x ,1 , x ,0 k e r A, x ,1 k e r A , x = x,0 + x,1 , x,0 k e r A, x,1 k e r A .
x

(1.27)

A A x + L L x = L g ,
(1.27)

A A x ,1 + L L x ,0 + L L x ,1 = L g .
0


(1.28)

,

A A x,1 = 0 . x,1 k e r A . (1.27), x,1 = 0 , ,

x = x,0 k e r A .
Q A = 0 , (1.28)

(1.29)

QL L x ,0 + QL L x ,1 = Q L g . QL L x,0 = Q L g QL L Qx,0 = Q L g .

(1.30)

(1.30) , (1.29),

(1.31)

g G ­ , (1.31) Q L L Q , L Q . .

181


:

2.

§1

. 1.6 f = 0 . 1.9 . 1.6. 1.1 1.2. 1.1. . ( S ; f , g

)

f F g G . -

. ( S ; f , g ) , , (1.1) f F . , A+ F . , A : H F . (1.1)

x = A+ f + Q y ,

(

Q = E - A+ A ,

)

(1.32)

y H ­ . , (1.1)

K = A+ f + Q y : y H .

{

}

(1.33)

( S ; f , g ) , (1.2) K g G . x (1.32) (1.2):

L Q y = g - L A+ f .


(1.34)

y = ( LQ

)

+

(

g - L A+ f + Q1 z ,

)

(

Q1 = E - ( L Q

)

+

LQ ,

)

(1.35)

z H ­ . y (1.35) (1.32), ( S ; f , g ) :

x = A+ f + Q ( L Q

)

+

(

g - L A+ f + Q Q1 z .

)

(1.36)

182


:

2.

§1

,

(S ; f , g )



. , (1.36) z H : Q Q1 z = 0 . Q Q1 = 0 . , 1.2, A L . , . . . . ( S ; f , g ) . A : H F , A
+

F : D A

()
+

=F.

(1.1) f F (1.32). x (1.32) (1.2), (1.34). , (1.2) (1.33) (1.34). 1.1 L Q : H G . (1.34) (1.35) ­ (1.36) . 1.2, (1.36), ( S ; f , g ) , (1.7). . 1.1 . 1.2. . (V ; f , g ) f F , g G > 0 x x ( f , g ) , 0 : l i m x = x , x H ­
0

. . f F , g G > 0

x ( f , g ) = x ( f , 0 ) + x ( 0, g ) ,

(1.37)

1.8 1.9 A L Q . 1.1 A L . .
183


:

2.

§2

. . (1.37) 1.8 1.9 x 0 . 1.6
0

li m x

(

f , g ) = x0 ( f , g ) E - ( L Q

(

)

+

L A+ f + ( L Q

)

)

+

g



x ( f , g ) - x0 ( f , g ) m
: 0 < < q0
-1 0

, q0 : 0 < q0 < 1 ­ ,

0 = (T ) ­ (1.8),
m= 1 T A A 1 - q0 0

(

)

+

L L T0 A+ f - E - L ( L Q

(

)

+

)

g

. 1.2 .

2. L - 2.1. . A : H F

L : H G ­ , f F g G ­ , H , F G ­ .

Ax = f ; Lx = g .
, § 1.2 :

(2.1) (2.2)

(S ; f , g )

:

(2.2) (2.1);

184


:

2.

§2

(V ; f , g ) : x H ,





( x)

= Ax - f

2

+ Lx - g

2

mi n ,

xH ,

(2.3)

> 0 , 0 . : ( S *; f , g ) : (2.1) (2.2);

(V *; f , g )

:

x H , (2.3), > 0 , . , , . 2.2. . , Q Q1 , § 1, :

QA = E - A+ A ,
QL = E - L+ L ,
, Q A , QL , QL
A

QLQ = E - ( L Q Q
AQ

A

)( )(
+

+

LQ

A

)

= E - ( A QL

A QL ) .

Q
AQ

Q

­ -

L , L QA A QL , . f 0 F , g 0 G ­ .

x0 = A+ f 0 + L Q

(

A

)(
+

g 0 - L A+ f

0

)

,

(2.4)

185


:
x0 = L+ g 0 + A QL

2.

§2

(

)(
+

f 0 - A L+ g
0

0

)

.

(2.5)

S *; f 0 , g

(

S ; f0 , g

)

-

(

0

)

(2.4) (2.5). ,

: 2.1. ) f 0 F g 0 G S ; f 0 , g

(

0

)

,

x = x0 + QAQLQ u ,
u H ­ . ) f 0 F g 0 G S *; f 0 , g

(2.6)

(

0

)



,
x = x0 + QLQAQ v ,

v H ­ . 2.1. ) f 0 F , g 0 G

(

S ; f0 , g

0

)

.

A x = f0 x = A+ f 0 + Q A y , yH .
(2.7)

L x = g 0

y L Q A y = g 0 - L A+ f 0 .
S ; f 0 , g (2.8)

(

0

)

, (2.8)

,

y = LQ

(

A

)(
+

g 0 - L A+ f
186

0

)

+ QLQ u ,

(2.9)


:

2.

§2

u H ­ . (2.7) ). ) ). 2.1 .

(

S ; f0 , g

0

)(

S *; f 0 , g

0

)

.

2.2. : )

Ax = f0 , . Lx = g0
f 0 F g 0 G ; ) :

(2.10)

f0 R ( A) ,

g 0 - L A+ f 0 R L Q

(

A

)

;

(2.11)

) :

g0 R ( L ) ,

f 0 - A L+ g 0 R A QL .

(

)

(2.12)

2.2. ) ), , (2.10) , ) ), ) ). ) ). f 0 F

g 0 G (2.10) .
(2.11) f 0 g 0 . (2.11) (2.10). (2.11). (2.10) (2.7). x (2.10), (2.8). (2.10) 187


:

2.

§2

(2.8). , (2.11). ) ) . ) ). (2.11). (2.10). (2.10) f 0 R ( A ) . (2.7) y (2.8). (2.11) (2.8) , , (2.10) . ) ) . 2.2 . 2.3. QAQL
Q

QLQ

AQ



k e r A k e r L ,

R QAQL

(

Q

)

= k e r A k e r L = R QL Q

(

AQ

)

.

(2.13)

2.3.

R QAQL

(

Q

)

ker A ker L .

(2.14)

x H ­ . y = QAQLQ x . A Q
A

L QAQL

Q

, A y = 0

L y = 0 . y k e r A k er L . (2.14) .


k e r A k e r L R QAQL

(

Q

)
(

.

(2.15)

y k er A k e r L ­ . QA y = y

L y = 0 .

QAQLQ y = Q

A

(

y - LQ

(

A

)

+

L QA y = Q A y - L Q

)

A

)

+

Ly = y .

188


:

2.

§2

, y R QAQL

(

Q

)

. (2.15) .

(2.13) . (2.13) . 2.3 . 2.4. ) ­ ) 2.2. u H v H ,
x0 + QLQA Q v = x0 + QAQLQ u ;

(2.16)

v H u H ,
x0 + QAQLQ u = x0 + QLQAQ v .

(2.17)

2.4. ) ­ ) 2.2 , , . (2.11) (2.7) (2.1). (2.2) (2.8). , (2.11), S ; f 0 , g

(

0

)
uH .
(2.18)

x = A+ f 0 + L Q

(

A

)(
+

g 0 - L A+ f

0

)

+ QAQL Qu ,

(2.12) (2.2) x = L+ g0 + QL y,

y H .

(2.1), A QL y = f 0 - A L+ g 0 . , (2.12),

(

S *; f 0 , g

0

)
x = L+ g 0 + A QL

(

)(
+

f 0 - A L+ g

0

)

+ QLQAQ v,

vH .

(2.19)

(2.18) (2.19) (2.10). 2.3 (2.16) (2.17).
189


:

2.

§2

2.4 . , 2.1 ­ 2.4 A L , . . ,

k e r A k e r L = {0} .

(2.20)

2.1. : ) (2.10) f 0 F g 0 G -

x ;
) S ; f 0 , g (2.4); )

(

0

)
0

x0 , -

(

S *; f 0 , g

)

x0 , -



(2.5); ) (2.11) (2.20); ) (2.12) (2.20);
x = x0 = x0 .

(2.21)

) ). ,

(

S ; f0 , g

0

)

,

(2.4). , S ; f 0 , g

(

0

)

-

(2.6). , u H (2.4) (2.10). (2.10) .

QAQL Qu = 0

(2.22)

u H . , (2.6) x0 , , S ; f 0 , g

(

0

)

-

, (2.4). ) ) .
190


:

2.

§2

) ). , S ; f 0 , g

(

0

)

(2.6). )

. (2.22). , (2.10) (2.18), u H ­ . (2.22) . ) ) . (2.21). ) ) ) ) , , (2.10) . ) ). (2.11) 2.2. (2.20). z k e r A k e r L . ) ) , x = x0 + z (2.10). (2.10) , z = 0 . (2.20). ) ) . ) ). , (2.20) (2.22). , x H A QA = 0 L QAQLQ = 0

QAQLQ x k er A,

QAQLQ x k e r L .

(2.11) (2.7), (2.9) (2.6). , (2.22), (2.6) (2.10). ) ) . ) ) ) ) , , (2.10) . 2.1 . 2.3. () .

(S ; f , g )

( S *; f , g ) f F g G

191


:

2.

§2

, :

f = A A+ f , 01 + g01 = L A f + L L Q

(

A

)(
+

g - L A+ f , f - A L+ g ,

)

,

(2.23)

f = A L+ g + A A Q 02 L + g02 = L L g ,

(

)(
+

)

.

(2.24)

. f F g G ­ , f 0 1 , g 01 , f f
01 02

,g

02

. ,

, , -

f F R ( A) R ( A) A . R ( A) , f
01

.

: 2.2. ) ( S ; f , g ) f F g G S ; f 0 1 , g

(

01

)

-

f

01

g

01

(2.23).

) ( S *; f , g ) f F

g G ( S *; f 02 , g g
02

02

)

f

02



(2.24). 2.2. ) .

( S ; f , g ) : 1) ; 2) ; 3) . ) ,

(

S ; f01 , g

01

)

f

01

g

01

(2.23)

192


:

2.

§2

: 1) ; 2) ; 3) . 1) ( S ; f , g ) f F g G . , 1.1) , 1.2) (2.1) (2.2) (2.1) . 1.1). (2.1) , (2.1) ,

f R ( A

)



R ( A) A . f D ( A+ ) A+ , ,

f

01

(2.23).

(

S ; f01 , g

01

)

.

1.2). f
+

A , (2.1)

x = A+ f + QA y .

(2.25)

y , (2.25),

L QA y = g - L A+ f .

(2.26)

(2.2) (2.1) f g , (2.26) . , g - L A+ f D ( L Q

(

A

)

+

)


01

(

LQ

A

)

+

. 01

, g .

(2.23). S ; f 0 1 , g

(

)

193


:

2.

§2

2) ( S ; f , g ) f F g G . , S ; f 0 1 , g

(

01

)

.

, (2.1) (2.25), (2.26):

y = ( LQ

A

)(

+

g - L A+ f ) + QLQ z,

zH .

(2.27)

( S ; f , g )

x = A+ f + ( L Q
f D ( A+ ) , f
01

A

)(

+

g - L A+ f ) + QAQL Q z ,

z H . (2.28)

R ( A) g - L A+ f D ( L Q

(

A

)

+

)

,

g 01 - L A+ f 0 1 R ( L Q

A

)

.

(

S ; f01 , g

01

)





x = A+ f 01 + ( L Q

A

)(

+

g 0 1 - L A+ f

01

)

+ QAQLQ w,
LQ
A

w H .

(2.29)

A+ A A+ = A+ L Q

(

A

) L(

+

)

+

= (LQ

A

)

+

, (2.30)

A+ f 01 = A+ A A+ f = A+ f ,

(
= ( LQ = ( LQ
A

LQ

A

)(

+

g 01 - L A+ f
A

01

)

=

)

+

(

L A+ f + L ( L Q

)(

+

g - L A+ f ) - L A+ f = LQ

)

A

)

+

(

L(LQ

A

)(

+

g - L A+ f

)) = (

A

)(

+

g - L A+ f ) . (2.31)

(2.30) (2.31) , (2.29) (2.28).

194


:

2.

§2

3) ( S ; f , g ) f F g G . z H (2.28) ( S ; f , g ) , QAQLQ = 0 . (2.29) S ; f 0 1 , g

(

01

)

. -

. ) . ) . 2.2 . 2.3. : ) f F g G ( S ; f , g ) : ) ; )

Ax = f01 , L x = g 01
; ) (2.20)

(2.32)

f01 R ( A) ,

g 01 - L A+ f 01 R L Q

(

A

) )

;

) (2.20)

g 01 R ( L ) ,

f 01 - A L+ g 01 R A QL .

(

2.3. ) ) 1.1, ) ) ) ) ) ) 2.1

f 0 = f 01 , g 0 = g 01 . ) ).

195


:

2.

§2

) ). . (2.32) f F g G . , -, A : H F . A+ F , , f (2.23). f
01

01



R ( A) , (2.32)

.

x = A+ f 01 + QA y ,

yH .

(2.32) y

L QA y = g 01 - L A+ f 01 .
, -, A L . 1.1 L QA : H G . L Q

(

A

)

+

G .



y = ( LQ


A

)(

+

g 0 1 - L A+ f

01

)

+ QLQ z ,

z H .

g 01 - L A+ f 0 1 = ( L Q

A

)(

LQ

A

)(

+

g - L A+ f ) R ( L Q

A

)

,

(2.33)

. , L QA y = g 01 - L A f
+ 01

.



y = ( LQ

A

)(

+

g 0 1 - L A+ f

01

)

+ QLQ z,

z H .

196


:

2.

§2

, (2.32) ,

x = A+ f 01 + ( L Q

A

)(

+

g 0 1 - L A+ f

01

)

+ QAQLQ z,

z H .

, QAQLQ z = 0 z H . , A QA L QA QL
Q

, w = QAQLQ z -

A , L . A L

w = QAQLQ z = 0 . , (2.32)
f F g G . ) ) . ) ). (2.32) f F g G . . (2.32) f F

g G (2.20). , x ­
(2.32), x + y y k er A k e r L . (2.32) , y = 0 . (2.20). (2.32) , A x = f . f
01 01

f F .

, A+ F , , A . x = A f
+ 01

+ QA y , y H

(2.32) y L QA y = g 01 - A f 0 1 . (2.32) , ,
+

g01 - A+ f 01 R ( L Q
197

A

)

.


:

2.

§2

(2.33)

g 01 - A+ f 0 1 = L ( L Q


A

)(

+

g - A+ f ) .

{

g - A+ f : f F , g G}
A

G , ( L Q
, L Q
A

)

+

.

. , -

A (2.20), A L . ) ) . 2.3 . 2.4. : ) f F g G S ; f , g : ) ; )

(

)

Ax = f02 , L x = g 02
; ) (2.20)

(2.34)

f02 R ( A) ,

g 02 - L A+ f 02 R L Q

(

A

) )

;

) (2.20)

g 02 R ( L ) ,

f 02 - A L+ g 02 R A QL .

(

2.4. 2.3, , , (2.32).

198


:

2.

§2

2.4. . ,

f F g G ( S ; f , g ) , ( S *; f , g )
(2.32) (2.34), A L . 2.5. f F g G ­ . (2.32) ( f 01 , g ) ( S ; f , g ) ; ) ( S *; f , g ) . 2.5. ) ­ ( S ; f , g ) : 1) ; 2) ; 3) . 1). ( S ; f , g ) f F g G . , ) (2.1) , ) (2.2) (2.1) . ). (2.1) ,
01

)

(2.23) :

f1 R ( A ) , f1 f R ( A ) .
f
A+ , , f
01

(2.23). -

(2.32) ( f 01 , g

01

)

(2.23) .

). f1 R ( A ) , (2.1) (2.25). y , (2.25), (2.26). (2.2) (2.1) f g , (2.26) . , g - A+ f -

199


:

2.

§2



(

LQ

A

)

+

, , g
01

01



(2.23). (2.32) ( f 01 , g .

)

(2.23) -

«» «» ) ) , (2.32) (2.23) , ( S ; f , g

(

f 01 , g

01

)



)



f F g G .
1) . 2).

(S ; f , g )

f F

g G . , (2.32)
. , (2.1) (2.25), (2.26) (2.27).

(S ; f , g )

(2.28).

, QAQLQ . x , (2.28), (2.32). , (2.32) . (2.32) . , z H

x = A+ f 01 + ( L Q

A

)(

+

g 0 1 - A+ f

01

)

+ QAQLQ z ,

zH

(2.32).

A+ f 0 1 = A+ f ,

(

LQ

A

)(

+

g 0 1 - A+ f

01

)=(

LQ

A

)(

+

g - A+ f ) ,

(2.29) (2.28) , , ( S ; f , g ) .

200


:

2.

§2

3).

(S ; f , g )

f F

g G . z H
(2.28)

(S ; f , g )

, QAQL Q = 0 .

(2.29) (2.32). «» «» . , (2.32) ( f 01 , g
01

)

(2.23) ( S ; f , g ) .

2.5 . 2.5 . 2.6. f F g G ­ . (2.34) ( f 02 , g ) ( S ; f , g ) ; ) ( S *; f , g ) . 2.6 2.5. 2.5. . . , , L A L . . 2.7. (V ; f , g
02

)

(2.24) :

)

f F ,

g G > 0 x ,

, ; : 0 < < <
x = ( A A + L L

)(

-1

A f + L g

)

(2.35)

201


:

2. § 3


x - x0 = M -1

,

(2.36)

x0 ­ S ; f , g , M > 0 ­ , . 2.7. = 1 / , (2.35)
x1 / = ( L L + A A) -1



(

)

(

L g + A f ) .

1.2 , , (2.36). 2.7 .

3. L - 3.1. . «» L «» , « »

(

A, L, f , g ) «-

f » A, L, , g .
- A : H F L : H G ­ , H , F G ­ . ,

(

)

A: H F , L: H G

{ A} , {L} { f } , { g}

­ ­ f F ,

g G , A- A µ , L-L µ ,

(µ

0) ,

(3.1)

202


:

2. § 3

- f f , f

g -g g ,

(

0) ,

(3.2)

µ > 0 , > 0 ­ , . , A , L (3.1)

A , L . , µ , ,
: O ( µ + ) ; ­ O ( µ + ) . , f 0 F g0 G

A x = f0 , Lx = g0

(3.3)

. f 0 g 0

0 F , g0 G , f - f f , f0 0 0 g0 - g0 g0 .
(3.4)

V ; , g 0 : x0 f0


(

)



0

( x)

= A x - 0 f

2

2 + Lx - g0 ,

xH

( 3 .5 )

> 0 (3.3).

203


:

2. § 3

. . , ,

=

m µ, q

(3.4)

m q ­ ,

m m0 ,

0 < q < 1,

(3.5) (3.6)

m0 = 2 (1 + A + L ) .


m1 =

1 1 + A+ + L L A 1 - q) (
m2 = 1+ L . (1 - q )

(

+

)

,

(3.7)

( 3 .8 )

. 3.1. (3.3) . , 0

x0 - x0 m1 + m2 µ x0 + m1 0 - f0 + m2 g 0 - g 0 , f

(

)

(3.9)

x0 ­ V ; , g 0 , x0 ­ f0
(3.3), m1 m2 (3.7) (3.8). 3.1 .3.3.

(

)

= P , = A A + E f f f

(

)

-1

f A , g = L + S g - L , f f

(

)

204


:

2. § 3



P = A A + E L = L Q ,

(

)

-1

A A ,

S = L L + E

(

)

-1

L L ,

L0 = L Q,

Q = A A + E

(

)

-1

(3.10)

,

> 0 . . 3.2. .

Ax - f

2

+ L x - g

2

mi n,

xH

(3.11)

x 0 x - x0 m3 + m4 µ + m5 ,
x0 ­ ( S ; f , g ) , m3 , m4 , m5 ­ , . 3.2 .3.3. 3.1 3.2 . 3.3. (3.3) . ,

f x0 - x0 m1 + m2 µ x0 + m1 0 - f0 + m2 g 0 - g 0 ,
f x0 ­ V ; , g 0 , x0 ­ 0

(

)

(

)

(3.3), m1 m ­ . 2


g = P g ,

g = L L + E

(

= Ag + S - A f f
205

(

) g )

-1

L g ,


:

2. § 3



P = L L + E A = A Q ,

(

)

-1

L L ,

S = A A + E

(

)

-1

A A ,

A0 = A QL ,

Q = L L + E

(

)

-1

,

> 0 . 3.4.

. .
2

A x - f

+ L x - g

2

mi n,

xH

x - x0 m3 + m4 µ + m5 , x x0 ­ S ; f , g , m3 , m , m5 ­ 4
. 3.2. . , .

(

)

R = A A + L L,

R = A A + L L .

3.1. .
- R 1 R - R

(

)

q,

(3.12)

- R 1

1 , (1 - q )

(3.13)

- R 1 A m1 , - R 1 A m2 ,
206

(3.14) (3.15)


:

2. § 3

q , m1 , m2 (3.5), (3.7), (3.8), ­ . 3.1. -

R > 0 .
R > 0 . , A L

R x x

x H .

, R 2 [41, . 204]
- R 1

1



.

(3.16)



R - R = A - A

(

)(

A - A + A - A A + A A - A +

)(

)

(

)

+ L - L ( L - L ) + L - L L + L ( L - L ) ,
(3.1) (3.4) : 0 < 1 ,

(

)

(

)

R - R m0 µ .
, (3.16) (3.4)

- R 1 R - R

(

)

- R

1

1 R - R hM =



1 h M = q . (3.17) hM q

(3.12) . (3.13). (3.17) 0 < q < 1 , [4 1 , . 206]

E + R

-1

(

R - R
207

)


:

2. § 3

E + R -1 R - R


(

)

-1



1 . 1- q
-1

(3.18)

- - - R 1 = E + R 1 R - R R

(

)

1

(3.18), (3.13). (3.13) . (3.14).

- - R 1 A = R 1 A A - A + A - A A
(3.13),

(

)(
1

)

-1

- A+ + A+ - R 1L L A

+

- - R 1 A R

A- A
1

(

A+ A

)

A+ +

- + A+ + R

L L A+
+



1 µ (1 + 2 A ) + µ (1 + 2 L ) + L L A+ + A (1 - q ) = mµ 1 A+ + A+ + L L A (1 - q ) (1 - q ) =
+

=

=

q 1 A+ + A+ + L L A+ = 1- q (1 - q ) = 1 + L L 1- q) (

(

)

A

+

= m1 .

(3.14) . (3.15).

- - - R 1 A = R 1 A - A + R 1 A ,
(3.13) (3.14) (3.15).
208

(

)


:

2. § 3

3.1 .

q n1 = 1 + (1 + 2 A ) A+ , n3 = n1 A+ , m n2 = q q + 1 + (1 + 2 A ) A+ , n4 = n2 , m m
+2

n5 = n1 A

+ n2 n3 , n6 = n2 n4 + 1 .

(

)

3.2. .

( (

A A + E A A + E

) )

-1

A n1 , A n2 ,

(

A A + E

)

-1

A n1 , A n2 ,

(3.19)

-1

(

A A + E

)

-1

(3.20) (3.21) (3.22)

P - A A+ n3 + n4 µ ,

(

A A + E

)

-1

A P - A+ n5 + n6 µ .

3.2. (3.19). A = A A A

()

+

,

(
+ A A + E

A A + E

)

-1

A = A A + E

(

)

-1

A A A
+

()
)(

+

=

= A A + E

(

)

-1

A A A

()
(

+.

(

)

-1

A A - A + A - A A - A - A A - A A

(

)(

)

)( )

+

.

209


:

2. § 3

(3.1), (3.4)

(


A A + E

)

-1



1



,

( )

A A + E

)

-1

A A 1 ,

(3.23)

(

A A + E

)

-1

A A A + E

(

-1

A A A+ + A A + E

(

)

-1



A - A A - A + A - A A + A A - A A+ + 1

(

) (A )
)

+





(

µ 2 + 2 A µ A+ A+ +

)

q (1 + 2 A m

A+ = n1 .

(3.19) . (3.19) . (3.20).

(

A A + E

)

-1

A = A A + E

(

)(
-1

A - A + A A + E

)(
-1

)

-1

A,

, (3.1), (3.19) (3.23),

(

A A + E

)

-1

A A A + E
-1

(

)

A- A +

+ A A + E

(

)

A

µ q + n1 = + n1 = n2 . m

(3.20) . (3.20) . (3.21).

P - A A+ = - A A + E + A A + E

(

)

-1

A A+ +
+

(

) A(
-1

A - A E - A A

)(

)

210


:

2. § 3

(3.19) (3.20),

P - A A+ A A + E + A A + E A A + E

(

)

-1

A A+ +

(

)

-1

A A - A E - A A+

(

)

-1

A A+ + µ A A + E

(

)

-1

A n1 A+ + µ n2 .

(3.21) . (3.22).

(

A A + E

)

-1

A P - A+ = - A A + E

(

)

-1

A A A

(

)

+

+

+ A A + E

(

)

-1

A P - A A

(

+

)+(

A - A A+ .

)

(3.1), (3.19), (3.20), (3.21),

(

A A + E

)

-1

A P - A+ = A A + E

(

)

-1

A

(

A A

)

+

+

+ A A + E

(

)

-1

A
2

(

P - A A+ + A - A A+

)

n1 A+

(

+ n2 n3 + n2 n4 + 1 µ .

)

(

)

(3.22) . 3.2 .

k1 = n1 A

+

L , k2 = n2 L + 1,

k q k3 = L+ + L+ 2 L0 + 1 k1 + 2 , 0 0 m k q k4 = k1 + k3 + 2 , k5 = k1k4 + k3 L+ , k6 = k2 k4 . 0 m

(

)

(3.24)

211


:

2. § 3

3.3. .

L - L0 k1 + k2 µ ,

(3.25) (3.26)

( (
(

L L + E L L + E

) )

-1

L0 k3 , L k4 ,

-1

(3.27) (3.28)

S - L0 L+ k5 + k6 µ , 0
k
i

i = 1, 6 (3.24).

)

3.3. (3.25).

Q - Q = A A + E

(

)

-1

A A

( ) +(
+

A A + E

)

-1

A A - A Q

(

)

(3.1), (3.19), (3.20) Q = 1 ,

Q - Q n1 A+ + n2 µ .


L - L0 L - L Q + L Q - Q n1 A
(3.25) . (3.26). (3.25)

+

L + n2 L + 1 µ .

(

)

L L - L0 L L - L Q + L Q - Q 0 n1 A
+

L + n2 L + 1 µ .

(

)

(3.29)

212


:

2. § 3

L0 = L Q : H G , L0 = L0 L L 0 0

()

+

. , (3.4), (3.29)

S 1 ,

(
L L + E

L L + E

)

-1

L0

(

)

-1

L0 L - L L 0

(L )

+ 0

+ S

(L )

+ 0



k q L+ + L+ 2 L0 + 1 k1 + 2 , 0 0 m

(

)

S (3.10). (3.26) .
(3.27).

(

L L + E

)

-1

L = L L + E

(

)(
-1

L - L0 + L L + E L

)(

)

-1

L0

(3.25) (3.26),

(
L L + E

L L + E

)

-1

(

)

-1

L - L0 + L L + E

(

)

-1

L0



1



(

k1 + k2 µ + k3 = k1 + k3 +

)

k2 q m

.

(3.27) . (3.28).

S - L0 L+ = 0 = L L + E

(

)

-1

L L - L 0

(

)(

E - L0 L+ - L L + E 0

)(

)

-1

L0 L+ 0

k4 k1 + k2 µ + k3 L+ = k1k4 + k3 L+ + k2 k4 µ . 0 0
213

(

)

(

)


:

2. § 3

(3.28) . 3.3 .

a1 = n3 , a2 = n4 , a3 = 1 ,
b1 = 2n5 + n2 A+ , b2 = 2n6 + 2 A+ , b3 = A+ ,

(3.30) (3.31)

c1 = ( L + 1) 2b1 + A+ k5 f + c2 = ( L + 1) 2b2 + A k6 + 2 A c3 = 2 ( L + 1) b3 f + g

(

(

)

)

+ k5 g ,
+

f +k g , 6

(3.32)

.

3.4. .

- f a +a µ +a f 0 1 2 3

(

)

f,

(3.33) (3.34) (3.35)

- A+ f b + b µ + b f 1 2 3

(

)

f,

g - g 0 c1 + c2 µ + c3 ,
ai , bi , ci

(

i = 1, 3 (3.30) ­ (3.32).

)

3.4. (3.30).

- f = P + AA f 0

(

+

)

f f +P - f . P 1,

(

)

, (3.1), (3.21)

- f n +n µ + f 0 3 4
(3.33) .
214

(

)

f.


:

2. § 3

(3.34). A+ = A A A

(

)

+

,

- A+ f = - A A + E f

(

)

-1

A A A

(

)

+

f+
+

+ A A + E

(

)

-1

A P - A A

(

+

)+(

A - A A+ + A f

)

(

- f . f

)

, (3.1), (3.20) (3.21),

- A+ f 2n + n A f 52

(

+

) + (

2n6 + 2 A+ µ + A+ f .

)

(3.34) . (3.35).

g - g 0 = S ( g - g ) + L A+ f - + L - A+ f + f f
+ S - L0 L+ 0

(

)(

)

(

)(

g - L A+ f + ( L - L ) A+ f + L0 L+ ( L - L ) A+ f . 0

)

, (3.1), (3.2), (3.28)

S 1 ,

g - g 0 ( L + 1) 2b1 + A+ k +

(

5

)

f + k5 g + f + k6 g µ +

(

(

L + 1) 2b2 + A+ k6 + 2 A

(

)

+

)

+ 2 ( L + 1) b3 f + g .
(3.35) . 3.4 . 3.3. 3.1. , A , L (3.3) .

x0 (3.3) .
215


:

2. § 3



- x0 - x0 = R 1 A 0 - f f - + R 1L g 0 - g

(

0

)-(

A - A x0 +

)

(

0

)-(

L - L ) x0 .

(3.1), (3.15) (3.17),

- 0 - x0 = R 1 A 0 - f0 + h x0 + x f - + R
1

L g 0 - g 0 + h x0

1+ L m2 0 - f 0 + f g -g + (1 - q ) 0 0
1+ L + m2 + (1 - q ) h x . 0

(3.7) (3.8) (3.9). 3.1 . 3.2. 2.3 ( S ; f , g

)

f F g G -

S ; f 0 , g

(

0

)

,

f 0 = A A+ f , g 0 = L0 L+ g - L A+ f + L A+ f . 0

(

)

(3.36)

(3.3), (3.36), . 3.1. x0 ­

(

S ; f , g ) . -

, x0
216

(

S ; f0 , g

0

)

.


:

2. § 3

A L .

- f - x = R 1 A* + L* g = R 1 A* f 0 + L* g
- f + R 1 A* - f

(

)

(

0

)

+

(

0

)

- + R 1L* g - g 0 ,

(

)

x - x0

- x - x0 = R 1 A* - f f - + R 1L* g - g

(

0

)+(

A - A x0 +

)

(

0

)+(

L - L ) x0 .

(3.1), (3.14), (3.15), (3.33), (3.35),

- x - x0 R 1 A* - + R

1

(

- f + A- A x + f 0 0

)

L*

(

g - g 0 + L - L x0 m3 + m4 µ + m5 ,
m3 = m2 a1 f + L +1 c, (1 - q ) 1
L +1 + m2 + x , (1 - q ) 0 L +1 c. (1 - q ) 3

)

m4 = m2 a2 f +

L +1 c (1 - q ) 2

m5 = m2 a3 f +

3.2 . 3.3 3.4 .

217


:

2. § 4

4. .. .. [104]. () Ax = f (4.1) ..

Ax - f

2

+ x mi n , x H ,
2

(4.2)

A : H F ­ , f F ­ , H F ­ . , A f µ : 0 < µ 1 > 0 ,

{ A}

­ A : H G

{f}

­

f F ,

A- A µ ,

- f f . f

( 4 .3 )

: 4.1. ) f F (4.1) , A ; ) f F (4.2) , 0 , A . , A . A A+ , k er A A ­ Q .
218


:

2. § 4

(4.1)

x = A+ f + Q y ,

( 4 .4 )

y H ­ , ­ ­

x0 = A+ f .

(4.5)

.. (4.2)

x = A A + E


(

)

-1

A f .

(4.6)

(

A A + E

)

-1



(

A A + E

)

-1

A -



( (

A A + E

)

-1

= -1Q +

( -1)
k =0 k =1



k

+ k A A

( (

)

k +1

,
k

(4.7)

A A + E

)

-1

A = A+ +

k ( -1)

k

A A + A+ .
-1 0

)

(4.8)

(4.7) (4.8) 0 < <

,

0 ­ A A .
x , (4.6) 0 , x0 ­ (4.1), (4.5), . 4.2.

(

)

+

x - x0 A

+3

.

(4.9)

,
219


:

2. § 4

(

A A + E

)

-1

A - A+ = - A A + E

(

)

-1

A A A

+

(

A A

)

+



(

A A + E

)

-1

A A 1,

A+ A A

(

)

+

A

+3

,

(4.9).

x = A A + E x = A A + E f = A A + E

( (

) )

-1

A f , A f ,

(4.10) (4.11) (4.12)

-1

(

)

-1

A A f .

: 4.3. = µ :

x - x0
f R ( A ) ;

=µ

m1µ + m2

(

)

f,

(4.13)

x - x0

=µ

m3µ + m4

(

)

f,

(4.14)

f R ( A ) , x x ­ (4.10) ­ (4.12).
4.3 . , § 1 § 2 , L = E g = 0 . mi

(

i = 1, 4 , (4.13) (4.14),

)

.. , .
220


:

2. § 4

mi .

(

i = 1, 4 -

)

O ( µ + [6 2 ].

)

-

= k µ 0 , k > 0 ­ , . [31] . . , . .

. .

, O µ

µ . , , : [16, 17 ­ 19, 20, 24, 26, 27, 32, 38, 40, 51, 60, 101, 104, 109, 113] . 4.1. = µ :

(

2/3

+

)

( (


A A + E A A + E

) )

-1

A n1 , A n2 ,

(

A A + E

)

-1

A n1 , A n2 ,

(4.15)

-1

(

A A + E

)

-1

(4.16)

n1 = 2 (1 + A

)

A+ ,

n2 = n1 + 1 .

4.1. (4.15). A , A = A A A+ .

(
+ A A + E

A A + E

)

-1

A = A A + E

(

)

-1

A A A+ +

(

)

-1

A A - A + A - A A - A - A

(

)(
1

)(

)(

A - A A+ .

)

,

(

A A + E

)

-1





,

(

A A + E

)

-1

A A 1

= µ ,
221


:

2. § 4

(

A A + E

)

-1

A A+ +

1



(

A µ + A µ + µ
+

2

)

A+

A+ + ( 2 A + µ ) A+ 2 ( A + 1) A

= n1 .

(4.15) . (4.15) .

(

A A + E

)

-1

A A A + E

(

)

-1

A - A + A A + E

(

)

-1

A ,

(4.3) (4.15), (4.16). (4.16) . 4.1 . 4.2. = µ :

( (


A A + E A A + E

) )

-1

A A - A A+ n3µ , A A - A+ A n3µ ,

(4.17)

-1

(4.18)

n3 = 2n2 + n1 A .

(

)

4.2. A ,

(

A A + E

)

-1

A A - A A+ = - A A + E

(

)

-1

A A+ +

+ A A + E

(

)

-1

A A - A - A - A A A+ .

(

)(
(

)

, (4.3), (4.15), (4.16) = µ ,

(

A A + E

)

-1

A A - A A

+

= A A + E
222

)

-1

A A+ +


:

2. § 4

+ A A + E

(

)

-1

A

(

A - A + A - A A A+ .

)

n1 A+ + 2µ n1 = n3 µ .
(4.17) . (4.18) . 4.2 . 4.3. f R ( A ) . f = A x0 , x0 (4.5). ,

x - x0 = A A + E

(

)

-1

A f - A+ f =

= A A + E

(

)

-1

A f - f - A - A A+ f -

(

)(

)

- A A + E

(

)

-1

A A A

(

)

+

f

, = µ 4.1,

x - x0 A A + E + A A + E
n1 A

+2

(

)

-1

A

(

A A

)

+

f+

(

)

-1

A

(

f - f + A - A A+ f
+

)



f + n2 f + µ A m1 = n1 A
+2

(

f

)=(

m1µ + m2

)

f,

+ n2 A+ ,

m2 = n2 .

(4.13) . (4.14). f R ( A ) .

223


:

2. § 4

x - x0 = A A + E

(

)

-1

A f - A+ f =

= A A + E

(

)

-1

A A - A A+ f + A A + E

(

)

(

)

-1

A A - A+ A A+ f +

+ A A + E + A A + E

(

)

-1

A A A + E

(

)

-1

A A f - f + A A - A A+ f .

(

)

(

)

-1

A A A + E

(

)

-1

, = µ , (4.3), 4.1 4.2,

x - x0 = A A + E + A A + E + A A + E + A A + E
n2 µ A

+

(

)

-1

A A - A A+ f +

(

)

-1

A A - A+ A A+ f +

(

)

-1

A

( (

A A + E

)

-1

A A f - f +

(

)

-1

A

A A + E

)

-1

A A - A A+ f

f + n3 µ A

+

f + n2 f + n2 n3 µ f = m3 µ + m4

(

)

f,

m3 = n2 + n3 A+ + n2 n3 ,
(4.14) . 4.3 .

(

)

m4 = n2 .

224


:

3.

§1

3
1. . , ,

(
­ ;

Ax ) ( t ) = f ( t

) )

( 1 .1 )

a x ( t ) + ( Ax ) ( t ) = f ( t

(1.2)

­ . : A=T + K (1.3) ­ ;

(T x ) ( t )

=

1 2



-



ct g

t-s x ( s ) ds 2

( 1 .4 )

­ , [64, . 42-43];

K = K1 + K

2

(1.5)

­ ;

( (

1 K1 x ( t ) = 2 1 K2 x (t ) = 2

)



-

k1 (


t - s ) x ( s ) ds

(1.6)

­ ;

)

-

k2 (

t + s ) x ( s ) ds

(1.7)

­ ;

k1 ( ) =

m=- M



M

bme

i m

,

k2 ( ) =

m= - M



M

cme

i m

(1.8)

­ ; 225


:

3.

§1 (1.9)

a, bm , cm

(

m = -M , . .., M

)

­ , M ­ ;

f (t ) H ,
H = H


x (t ) H



(1.10)

­ , ;

[

- , ] = f ( t ) : f ( t + ) - f ( t ) H

{

, f





}

(1.11)

­ 2 - ;

f
­ H ;

H

=f

C

+H

, f

(1.12)

f

C

= ma x f ( t
- t

)
)

(1.13)

­ C = C [ 0, 2 ] ;

H

, f

=

- t s

sup

f (t ) - f ( s t-s


(1.14)

­ f ( t ) ;

f (t ) - f ( s ) H
­ ;

, f

t-s



(1.15)

L2 = L2 [ - , ] = f ( t ) :



-

f (t )

2

dt < +

(1.16)

­ 2 - ;

f
­ L2 .

L2

=



-

f (t )

2

dt

(1.17)

226


:

3.

§2

2. 2.1. . f ( t ­ [ - , ] .

)

fk =

1 2



-



e

- i kt

f (t ) dt,

k .

(2.1)

f k f ( t ) ,

k = -





fk e

ikt

(2.2)

­ f ( t ) , (2.2) ,


f (t ) =

k = -




fk e

ikt

,

(2.3)

,

f (t )

k = -



fk e

ikt

.

(2.4)

, f ( t 2 .

)



n = 0, 1, 2, . . .

(

Sn f

)

(t ) =

n

k =- n

fk e

ikt

,

(2.5)

.

227


:

3.

§2

. Sn f (2.1):

(

)

(t )

f

k

(

Sn f

)

(

t) =

k =- n



n

1 2



-



e

-iks

f ( s ) ds e

i kt

=

1 2



-

f ( s ) kn =-

n

ik t -s e ( ) d s . (2.6)

. i t -s e ( ) 1 , ,

t - s 2m , m ,

k =- n
-i t -s 2



n

ik t - s - in t - s -i e ( ) =e ( ) +e (

n -1)( t - s

)

in t - s + ... + e ( ) =

=

e

i( n +1)( t - s

)

i t -s e ( ) -1

-e

- i n( t - s

)

=

e

(
e

i e(

n+1)( t - s

)

-in t - s -e ( )

-i

t -s 2

)

(

e

i( t -s

)

-1

)

=

=

e

1 i n + ( t - s 2 i

)

-e -e

1 - i n + ( t - s 2 -i t -s 2

)

e =

t -s 2

=

sin

(

2n + 1) ( t - s ) ( 2n + 1) ( s - t ) sin 2 2 = , x-t s -t sin sin 2 2



k =- n



n

ik t - s e ( )=

si n

(

2n + 1) ( s - t ) 2 . s -t si n 2

( 2 .7 )

t - s = 2m , m , e

i( t - s

)

= 1 , ,
k =- n

k =- n



n

ik t - s e ( )=



n

1 = 2n + 1 .

228


:

3.

§2

(2.7) t - s 2m , m . (2.7) t - s = 2m , m , , t - s 2m , m . (2.6) (2.7),

(

Sn f

)

(t )

=

1 2



-

f (s)

si n

(

2n + 1) ( s - t ) 2 ds . s -t sin 2

s - t = z 2 - f ( t ) ,

(


Sn f

)

(t )

=

1 2



-

f(

t+z

)

sin

(

2n + 1) z 2 dz . z si n 2

(2.8)

1 Dn ( z ) = 2

si n

(

2n + 1) z 2 z sin 2

(2.9)

. (2.5) :

( m f )

(t )

=

1 m

m -1 n=0

(

Sn f

)

( t ),

m ,

(2.10)

.
m -1 n =0



2 mz 2n + 1) z s i n 2 sin = , z 2 sin 2

(

229


: :

3.

§2

( m f )
=

(t )

=

1 m


m-1



n =0

1 2



-

f(

t+z

)

sin

(

2n + 1) z 2 dz = z sin 2

1 2m

-



f (t + z ) z si n 2
2

m -1 n =0



sin

(

2n + 1) z dz = 2

=

1 2m



-



mz si n 2 z si n 2

f ( t + z ) dz,

m .

(2.11)

,



m

(

mz sin 1 2 z) = 2m s i n z 2



2

(2.12)

­ . ,

(

Sn f

)

(t ) = (
t) =




-

Dn ( z ) f ( t + z ) d z ,

(2.13)

( n f

)

-



n

(z) f (

t + z ) dz .

(2.14)

(2.9) (2.12). 2.1. n = 0, 1, 2, . . .


-



Dn ( z ) d z = 1 .
230


: (2.7). 2.2. n
n

3.

§2

(t )

(2.15)

n ( -t ) = n ( t ) .

2.3. x [ - , ] n

n (t ) 0 .

(2.16)

(2.15) (2.16) ­ (2.12). 2.4. n


-



n ( t ) dt = 1 .

(2.17)

f ( t ) 1 ,

1, fk = 0,
Sn f



k = 0, k 0.

(

)

(t )

1 t [ - , ] , n = 0, 1, 2, . . . , ,

( n f )

(t )

1 t [ - , ] , n . (2.14)

(2.17). . 2.5. : 0 < <
n n

li m ( ) = 0 ,

(2.18)



n ( ) n ( t ) d t =




- -



n ( t ) dt .

(2.19)

231


: ,

3.

§2

t ,

sin

t . 22
nt sin 2 t si n 2 2 1 2 = . 2 2
2

(2.20)

(2.12) (2.20) (2.18):




nt sin 1 n ( t ) dt = 2 2n s i n t 2

2

dt

2



2 ( - ) 1 2 dt = n 2 0 n . 2n

. 2.2. . , , . , , , - : , , , , . [43, . 399]. . f ( t


)

­ .

t

-



f ( x + t ) - f ( x) dt t

- > 0 , Sn f

f ( t ) t f ( t ) .
232


:

3.

§2

, t . f ( t

)

­ 2 - , -

. f ( t

)

-

, t

(

Sn f

)

(t )



1 ( f (t + 0) + f (t - 0 2

)) (

n ) .

t f ( t ) ,

f ( x + 0) = f ( x - 0) = f ( x
, ,

)

(

Sn f

)

( x)

f (x

)(

n ) .

. , . . , f ( t ) , , . , f ( t

)

­

(, , ). , . f ( t > 0 ,


)



t E [ - , ] , > 0

-



f (t + s ) - f (t s

)

ds < , t E .

233


: f ( t E .

3.

§2

)



2.3. . . , , f1 ( t

)

f

2

(t )

­ ,

:

f1k = f 2 k , k ,


(2.21)

f1 ( t ) f 2 ( t ) .
, f1 ( t

(2.22)

)

f

2

(t )



L2 .
k

(t )

= ei kt , k ,

f = f1 - f 2 :

f
(2.21)

2

L2

=

k =-





f

2

k

.

(2.23)

f k = f1k - f

2k

= 0,

k .

(2.23)

f

L2

=0.

(2.24)

(2.24) , f ( t ) . f ( t

)

-

, : f ( t ) 0 . (2.22). .

234


:

3.

§2

,

f (t

)

,

. [43, . 405] . . f ( t

)

­ 2 -

, n f

{

}



f ( t ) . . n f f ( t

)

n . , -

f ( x

)

-

. M > 0 ,

t [ - , ]

f (t ) M ,

t [ - , ] .

(2.25)

, > 0 > 0 , t - t <

f ( t ) - f ( t ) <


2

(

t - t < ) .

(2.26)



f (t ) - n f

(

)

(t ) =



-

f (t ) - f (t + s )

n

(s)

ds .

(2.27)

:

J- =

- -



f (t ) - f (t + s )

n

( s)

ds ,

(2.28)

235


:

3.

§2

J0 =



-




f (t ) - f (t + s ) n ( s ) ds ,

(2.29)

J + = f (t ) - f (t + s )
(2.19), (2.25), (2.28) (2.30)



n

(s)

ds .

(2.30)

J - 2 M n ( ) ,

J + 2M n ( ) .

(2.31)

n0 , n n0 > 0

2 M n ( ) <
(2.26) (2.29)


4

.

(2.32)

J0 <



-

f (t )

- f (t + s )

n

(s)

ds <


2



-



n ( t ) dt


2



-



n (t ) dt =


2

.

(2.33)

(2.27) (2.31) ­ (2.33),

f ( x) - n f

(

)

(t )

J- + J0 + J+ <


4

+


2

+


4

=

t [ - , ] . , > 0 , . .

236


:

3.

§3

3. 3.1. . 2 - , ,
k

(t )

= ei k t , k ,

. , . . 3.1. f ( t ) H


[

0, 2 ] .

f ( t )
H
0 0 < t - s <

[

0, 2 ] , -

li m s up

f (t ) - f ( s t-s


)

=0. f (t

( 3 .1 )

. .


)



H

[

0, 2 ] . > 0 -

g ( t ) ,

f -g

H [ 0 , 2

]

<


2

.

(3.2)

g ( t

)

-

K > 0 ,

g ( t ) - g ( t) K t - t

237

- t t .

(3.3)


:

3.

§3

> 0 > 0 ,

K

1-

<


2

.

(3.4)

- t t , t - t < , (3.2) ­ (3.4),

f ( t ) - f ( t t - t


)



f ( t ) - g ( t ) - f ( t) + g ( t t - t
f -g


)

+

g (t ) - g (t t - t


)



H [ 0 , 2

]

+

K t - t t - t
1-



=

= f -g

H [ 0 , 2

]

+ K t - t

<


2

+


2

= .

(3.5)

, (3.5),

0 0 < t - t <

li m

s up

f (t ) - f (t t - t


)

,

(3.6)

> 0 . , (3.6) (3.1). . . f ( t
-

)

(3.1).

f ( t ) =



( t - ) f ( s ) d s,
-

>0,

(3.7)



( u ) = c e 0,


2

2 2

-u

,



u < , u ,

238


: c
-

3.

§3



( u ) d u = 1 :
-1

(3.8)

c =

-



2 exp - 2 du . 2 -u

( 3 .9 )

u = v , , (3.9) > 0 :



-



2 exp - 2 du 2 -u


-1

=

1 1 1 dv . exp - 2 -1 1- v

f

(t )

, (3.7),

. ( s = t + u f


)

(t )



f ( t ) =

-



( u ) f ( u + t ) d u ,

> 0.



f - f

H [ 0, 2

]

0



0.

(3.10)

, (3.8),

f ( t ) - f ( t ) =

-



( u ) f ( u + t ) - f ( t ) d u .



f ( t ) - f ( t )

-

-



( u ) f ( u + t ) - f ( t ) d u




( u ) d u s u p f ( u + t ) - f ( t ) H
- t u <

( f )



.

(3.11)

239


: :

3.

§3

f ( t ) - f ( t ) - f ( t ) + f ( t t - t


)




-

-



( u

)

f ( u + t ) - f ( t ) - f ( u + t) + f ( t t - t


)

du





( u ) d u

- t t u <

sup

f ( u + t ) - f ( t ) - f ( u + t) + f ( t t - t


)

.

(3.12)

( (3.1)) , > 0 > 0 , t - t <

f ( t ) - f ( t) <


2

t - t .



(3.13)

t - t < . (3.13) u

f ( u + t ) - f ( u + t) <



2

t - t ,



f ( u + t ) - f ( t ) - f ( u + t) + f ( t) f ( u + t ) - f ( u + t ) + f ( t ) - f ( t) < t - t .
(3.12), (3.14),


(3.14)

f ( t ) - f ( t ) - f ( t ) + f ( t t - t


)

sup
t t

t - t
t - t





= .

(3.15)

t - t . 0 > 0
0 =


2

.

(3.16)

240


: (3.11) 0 < < 0

3.

§3

f ( t ) - f ( t ) - f ( t ) + f ( t t - t


)





f ( t ) - f ( t t - t < 2


)

+


f ( t ) - f ( t t - t


)

<

H





( f )



= 2 2








= .

(3.17)

, 0 < < 0 , t t , (3.15) (3.17)

f ( t ) - f ( t ) - f ( t ) + f ( t t - t
0 < < 0


)

< .

(3.18)

H



(

f - f < ,

)

(

0< <

0

)

.

(3.19)

(3.11), (3.16), (3.18) (3.19)

f - f

H [ 0 , 2

]

<



0 < < 0 .

, > 0 , (3.10). 3.1 . 3.2. . H


[

0, 2 ]


. 3.1. f ( t ) H n f

[

0, 2 ] . -

{

}

­ f ( t 241

)


: H (3.1).


3.

§3

[

0, 2

]

-

. . H


{ n f }

f ( t

)

-

[

0, 2 ] ,

f -n f

H [ 0, 2

]

0



n.

(3.20)

(3.1). . 3.1 (3.1). . . (3.1). (3.20). > 0 ­ . f ( t n0 , n n0

)

-

2 - ,

2 t t
H

f -n f



[

0, 2

]

<



.

(3.21)

f (t ) - n f

(

)

(t )

- f ( t) + n f


(

)
f

( t)

t - t





f ( t ) - f ( t t - t


)

+

(

n f

)

( t ) - (
t - t


n

)

( t)

.

(3.22)

(3.1) > 0 > 0 , t - t < x

f ( x + t ) - f ( x + t t - t


) <

4

x .

(3.23)

242


:

3.

§4



( n f (
n f

)

(t ) = f (
- n



x + t )

n

( x)

d x , (3.22) (3.23)

)

( t ) - (
t -t'



f

)

( t ')



-



f (x +t)- f (x +t' t -t'


)



n

( x)
.

dx



-





n

( x)

dx sup

f (x + t) - f (x + t ' t -t'


- x 0 < t -t ' <

) <

4

(3.24)

(3.21), (3.23) (3.24),

f -n f

H

<


2

+


4

+


4

= ,

. 3.1 .

4. f ( t ) H . f ( t )

f (t ) =



k = -





fk e

ikt

,

(4.1)

fk =

1 2

-



e

- i ks

f ( s ) d s,

k

(4.2)

­ f ( t ) . (4.1)

g g

n, +

(t ) =
=



m=n n

fk e

ikt

,

(4.3)

n, -

(t )

m= -



fk e

i kt

(4.4)

, n ­ . 243


:

3.

§4

H . 4.1. f ( t ) H




-

­ .

n (4.3) (4.4) :

g

n, +

(t )

H ,

g

n, -

(t )

H .

(4.5)

4.1. f ( t [43, . 403]: E [ - ,

)

. -

]

-

f ( t ) > 0 > 0 ,


-



f (t + ) - f (t

)



d <

t E , f ( t ) E . . f ( t

)

,

[ - ,

]

, ,

. . > 0 ­ . 0 < <



2H
, f

, H

, f

­

f ( t ) H .

244


:


3.

§4

-



f (t + ) - f (t

)



d



-



H

, f







d =

2H

, f



<

t [ - , ] . . , . , (4.1) f ( t [ - , ] , (4.1). n = 0 :

)

g
= e
it

0, +

H .

(4.6)



( ) = f ( t ) , (1.5) ( ) =

k =-





f k k .

( 4 .7 )

( ) H . ­ [64, . 58] ,

(z) =

1 2 i

=1



( ) d , -z

z <1

(4.8)

z < 1 : 0 < < 1
+

(z)

-

z = 1 , :



+

(

( z ) , z ) = li m ( w ) , z w 1 w<



z < 1, z = 1.

245


:

3.

§4

z = 1 , ,




+

(z) =

k =0

fk z k ,

z =1.

(4.9)

z : z < 1 = 1

1 1 = -z
(4.7) (4.10) (4.8):

m =0





z

m m


m m

.

(4.10)

(z) =

1 2 i

=1



1



m =0





z



k =-





f k k d .

(4.11)

(4.7) (4.10) : = 1 , , (4.11) :

(z) =

1 2 i

m=0





z

m

k =-





f

k

=1





k - m-1

d .

(4.12)

= eit ,

1 2 i

=1





k - m -1

d =

1 2



-



i e(

k - m )t

0, dt = 1,



k m, . k m.

(4.13)

(4.12) (4.13)

(z) =





k =0

fk z k ,

z <1.

, (4.7) ( = eit )



0, +

( ) = f k
k =0



k

,

=1

(4.14)

246


:

3.

§4

= 1 . , > 0 N ( ) ,

n N ( ) m = 0, 1, 2, . . . ,
n+m k =n



fk

k

< .

(4.15)

, (1.5) , > 0 N1 ( ) , n N1 ( m1 , m2 = 0, 1, 2, . . . ,

)

f (t ) -

n+ m2


(

k = - n + m1

)

fk e

ikt

<


2

(4.16)

t [ - , ] . N ( ) = N1 ( ) + 1 . n N (

)

m = 0, 1, 2, . . . , (4.16),

f (t ) -

k =- n



n -1

fk e

i kt

<


2

,

f (t ) -

n+m k =- n



fk e

i kt

<


2

.

= eit
n+m k =n



f k

k

=

n+m k =n



fk e

ikt

=

n+m k =- n



fk e

ikt

- f (t ) -

k =- n


<

n -1

fk e +

i kt

+ f (t ) = .

f (t ) -

n+m k =- n



fk e

ikt

+ f (t ) -

k =- n



n -1

fk e

i kt


2


2

(4.15) . , (4.14) = 1 .



0

( z) =



k =0

fk z k ,

z 1.

(4.17)

247


:

3.

§4

z 1 . z = 0 (4.17) . z 0 ,

=

z , z

ak = f k k ,

bk = z ,

k

S

n, k

=


l =0

k

a

n+l

,

(4.18)

(4.17)



0

( z) =



k =0

ak bk .

(4.19)

(4.19) [34, . 15]. > 0 ­ . (4.14) = 1 N n N
2 2

( )

,

( )

p = 1, 2, . . . ,
n+ p k =n



f k

k

<


2

,

(4.20)

(4.18)

S

n, k

<, 2



k = 0, 1, 2, . . ., p .

(4.21)

:



p

k =1

an+ k bn

+k

=


S

p

k =1

(

S

n, k

-S

n , k -1

)

bn

+k

=


k =1

p

S n, k bn

+k

-


k =1
+1

p

S

n, k -1 n+ k

b

=

=



p -1 n, k

k =1

(

bn

+k

- bn - bn

+ k +1

)

+S

n , p n+ p

b

- Sn, 0bn +S







p -1

S

k =1

n, k

(

bn

+k

+ k +1

)

+S

n, p

bn

+p

n, 0

bn+1

248


:

3.

§4



(
k =1

p -1

bn

+k

- bn

+ k +1

)

+ bn

+p

+ bn +1 ma x S 0 k p

n, k

= 2bn +1 ma x S
0 k p

n, k

. (4.22)

z 1 , bn

+1

=z

n +1

1 . (4.20) ­ (4.22) , < 2bn + k



p

k =1

an + k bn


2

+k

.

(4.23)

(4.23) ,

k = n +1



p

fk z k < ,

z 1

n N

2

( )

p = 1, 2, . . . . (4.17)

z 1 . (4.6) ,
+

(z)

, z 1 ( z ) ,

z < 1 . , z < 1

0

(z)

= ( z ) . -



+

(z)



0

(z)

,



0

(z)

= (z) ,

z 1.

­ [64] (4.6). , 1.1 n = 0 . n ­ .
n-1

g

n, +

(t ) (t )

=g =g

0, +

(t ) -

k =1
-1

f k ei k t , f k ei k t ,



n>0, n<0.

(4.24)

g

n, +

0, +

(t ) +



(4.25)

k =- n

249


: g
0, +

3.

§5

(t )

H ,
ikt


k =1

n-1

fk e

(

n>0

)



k =- n



-1

fk e

i kt

(

n<0

)


(4.26)

H . , (4.24) ­ (4.26), g
n, +

(t )

H



n .
g
n, -

(t )

H



n -

g

n, -

(t )

= f (t ) - g

n +1, +

(t )

,

n ,


H . 4.1. .



5. , A a E + A , (1.3) ­ (1.9) . 5.1.


x (t )

k =-



xk e

i kt

,

(5.1)



(

Ax ) ( t ) i

- M -1 k =-



xk e

ikt

+

k =- M

(

-1

i + bk xk + ck x- k e

)

ikt

+ b0 + c0 x0 +
i kt

(

)

+



M

k =1

c x + -i + b x e k k k -k
250

(

)

i kt

-i

k = M +1





xk e

.

(5.2)


:

3.

§5

5.1. (5.2) .

(T x ) ( t )


i

k =-



-1

xk e

ikt

-i





k =1

xk e

ikt

.

(5.3)

rk =
,

1 2



-



u c t g ei k u d u , 2

k ,

(5.4)

rk = i s i g n k ,

k .

(5.5)

,

u u c t g ei u = c t g + i + i e i u . 2 2
(5.6) e
i( k -1)u

(5.6)

:
k -1)u i + ie ( k -1)u

u ct g e 2

iku

ui = ct g e ( 2

+ ie

iku

,

u [ - , (5.4)


]



rk = rk -1 +
,

i 2


-



i e(

k -1)u

du +

i 2

-



ei k u d u ,

k .

(5.7)

1 2

-



e

imu

1, du = 0,



m = 0, m 0.

.

(5.8)

(5.4) k = 0

r0 =

1 2



-



u ct g d u = 0 . 2

(5.9)

251


: (5.7) ­ (5.9)

3.

§5

rk = i,

k 1;

rk = -i,

k -1 .

(5.10)

(5.5). t - s = u (5.10),

1 2



-



ct g

t - s iks 1 e ds = - e 2 2


ikt

-



u c t g e i k u d u = -i e i k t s i g n k . 2

, T (5.1),

(
=

Tx ) (t ) 1 2

1 2



-



ct g

t-s 2

k = -





xk eiks d s =

k = -







xk e

ikt

-



u c t g e i k u d u = -i 2

k =-





xk eikt s i g n k .

(5.11)

(5.3) . ,

(

K1 x ( t ) =

)

1 2



-



k1 ( t - s ) x ( s ) d s =

1 2



-

k M =-

M

ik t -s bk e ( )

m= -





xme

ims

ds =

k =- M



M

bk


m= -





xm e

i mt

1 2



-



-i e(

k -m )s

ds =
M

k =- M



M

bk xk e
)


ikt

,

(5.12)

(

K2 x (t ) =

)

1 2

-



k2 ( t + s ) x ( s ) d s =

1 2



-

k =- M
ds =

ik ck e (

t +s

m=-



xm e

i ms

ds =

k =- M



M

ck

m= -





xm e

i mt

1 2



-



i e(

k +m)s

k =- M



M

ck x- k e

ikt

.

(5.13)

(5.11) ­ (5.13), (5.2). 5.1 . 252


: 5.1.

3.

§5

x (t ) =


k =-





xk e

ikt

.

(5.14)

a x ( t ) + ( Ax ) ( t ) = ( a + i + +

)

- M -1 k =-

xk e

ikt

+

k =- M

(

-1

a + i + bk xk + ck x- k e

)

ikt

+ a + b0 + c0 x0 +

(

)



M

k =1

c x + a - i + b x e k k k -k

(

)

ikt

+ (a - i

)



k = M +1

xk e

ikt

;

(5.15)

, : ) x ( t ) H , a x ( t ) + ( A x ) ( t ) H ; ) x ( t ) L2 , a x ( t ) + ( A x ) ( t ) L2 . . ) x ( t ) (5.14) H ,
- M -1

4.1

k = -



xk e

ikt



k = M +1





xk e

ikt

-

H . , (5.15) H . .

ax (t ) + ( Ax ) ( t ) H .
) x ( t ) L2 .
- M -1 k =-



xk e

ikt


2

k = M +1





xk e

i kt

- M -1

L2 ,

k =-



xk



2

k = M +1





xk

2



k =-



xk

­



253


:

3.

§6

x ( t ) L2 . , (5.15) L2 , L2 . 5.1 .

6. (1.1) (1.2)

i xk = f k , i + bk xk + ck x- k = f k , b0 + c0 x0 = f 0 , ck x- k + -i + bk xk = f k , -i xk = f k ,

- < k - M - 1, - M k - 1, k = 0, 1 k M, M + 1 k < + , - < k - M - 1, - M k -1, k = 0, 1 k M, M + 1 k < + ,
(6.2) (6.1)

( (

)

) (

)

( a + a+ a+ ck x- ( a -

( (

i ) xk = f k , i + bk xk + ck x- k = f k , b0 + c0 x0 = f 0 ,
k

)

i ) xk = f k ,

+ a - i + bk xk = f k ,

(

)

)

(1.1) (1.2). ,

Ax = f ,

f= f

( k)

k = -

(6.3)

254


:


3.

§6

-

k ( t, s ) x ( s )

ds = f ( t ) ,

(6.4)



( fk )
k = -

k = -

­

f ( t ) ,

( xk )

(6.3) -

x ( t

)

(6.4) , ,

x ( t ) xk

()

k = -

.

6.1. ) (6.1) (1.1); ) (6.2) (1.2). 6.1. ), (1.1) (6.1) (1.2) (6.2) a = 0 . x ( t

)

(1.2)

f ( t ) .

( xk )

k = -



( fk )

k = -

:

x (t ) =
- M -1

k =-





xk e

ikt

,

f (t ) =

k = -





fk e

ikt

.

(6.5)

(6.5) (1.2) , 5.1,

(

a+i

)

k = -

xk e

i kt

+

k =- M

(
M k =1


-1

a + i + bk xk + ck x- k e

)

ikt

+

+ a + b0 + c0 x0 + + (a - i

(

)



c x + a - i + b x e k k k -k
xk e
ikt

(

)

i kt

+
( 6 .6 )

)



k = M +1

k =-





fk e

i kt

.

255


: , xk (6.2)

3.

§6

()

k = -

-

( fk )

k = -

. ,

( xk )

k = -

(6.2)

( fk )

k = -

, (6.6).

, x ( t ) (1.2) f ( t ) . 6.1 . (1.1) (1.2)

A: H H , aE + A : H H ,
:

A : L2 L2 , a E + A : L2 L2 .

( 6 .7 ) (6.8)



L2

( A) H ( A) (
a E + A)
H

­ (6.7),



L2

(

a E + A ) ­ (6.8),

0 = ±i, b0 + c0 , 0, k : k = 1, 2, . . ., M , a = a ± i, a + b0 + c0 , a , k : k = 1, 2, . . ., M , 0, k = b- k + bk ± 2

1 1

{

}

{

}

(

b- k - bk + i 2

)

2

+ 4c- k ck ,

a , k = 2a + b- k + bk ± 2


(

b- k - bk + i 2

)

2

+ 4c- k ck .

(6.7) (6.8) .

256


: 6.2.

3.

§6


L2

L2

( A) = H ( A) =
H

0

,

(6.9) (6.10)

(

a E + A) =
0

(

aE + A) = a ;



a

A ,

­


a E + A H

L2 .

6.2.

Ax (t ) - x (t ) = f (t ) ,

(6.11)

. (6.5) (6.11),

( i - ) xk = i + bk - b0 + c0 - ck x- k + -i - ( i + ) xk

( (

)

fk , xk + ck x- k = f k ,

- < k - M - 1, - M k -1,
k = 0,
(6.12)

(

)

x0 = f 0 ,

+ bk - xk = f k , = fk ,

)

1 k M,
M + 1 k < + .

0 , (1.1) f ( t ) H


f ( t ) L2 ,

H



L2 . 0 .

±i ,

b0 + c0 ,

k i + b- k - -i + bk - - c- k ck 0

(

)(

)

257


:

3.

§6

, , (6.12) x ( t ) (6.11):

xk =
xk =

1 f, i- k

- < k -M - 1 ,
k = - M , . . ., - 1 , k =0,

(6.13)

1 -i + b- k - f k - ck f - k , k x0 = 1 f, b0 + c0 - 0

(

)

(6.14)

(6.15)

xk =

1 -c f k k

-k

+ i + b- k - f k ,

(

)

k = 1, . . . , M ,

(6.16)

xk = -

1 f, i+ k

M + 1 k < + .

(6.17)

f ( t ) H , (6.13) ­ (6.17) ) 5.1 x ( t ) H ; f ( t ) L2 , ) x ( t ) L2 . 0 . , A . x ( t ) , : 1) = i , x ( t ) = e
ikt

k : - < k - M - 1 ; k : M + 1 k < + ;

2) = -i , x ( t ) = e

ikt

3) = b0 + c0 , x ( t ) = C = co n s t ; 4) =

1 b + b + 2 -k k

(

2 b- k - bk + i 2 + 4c- k ck ,

)

258


:

3.

§6

x ( t ) = 2c- k e


- ikt

- b- k - bk + i 2 -

(

b- k - bk + i 2

)

2

+ 4c- k ck e

ikt

c- k + b- k - bk + i 2 - x ( t ) = bk - b- k - i 2 -

(

b- k - bk + i 2

)

2

+ 4c- k ck 0 ,
- ikt

(

2 b- k - bk + i 2 + 4c- k ck e

)

- 2ck e

ikt



c- k + b- k - bk + i 2 - ck + bk - b- k - i 2 -

(

b- k - bk + i 2

)

2

+ 4c- k ck = 0 , + 4c- k ck 0 ,

( (

b- k - bk + i 2

)

2

x ( t ) = C1e


- i kt

+ C2eikt , C1 C2 ­ b- k - bk + i 2

c- k + b- k - bk + i 2 - ck + bk - b- k - i 2 -
5) =

)

2

+ 4c- k ck = 0 , + 4c- k ck = 0 ;

( (

b- k - bk + i 2

)

2

1 2

b- k + bk -

b- k - bk + i 2

)

2

+ 4c- k ck ,

x ( t ) = 2c- k e


- ikt

- b- k - bk + i 2 +

(

2 b- k - bk + i 2 + 4c- k ck e

)

ikt

c- k + b- k - bk + i 2 +
x ( t ) = bk - b- k - i 2 +

(

b- k - bk + i 2 + 4c- k ck 0 ,

)

2

(

b- k - bk + i 2

)

2

+ 4c- k ck e

- ikt

- 2ck e

ikt

259


:

3.

§6



c- k + b- k - bk + i 2 + ck + bk - b- k - i 2 +

( ( (

b- k - bk + i 2 + 4c- k ck = 0 ,

)

2

b- k - bk + i 2

)

2

+ 4c- k ck 0 ,

x ( t ) = C1e


- i kt

+ C2eikt , C1 C2 ­ b- k - bk + i 2 + 4c- k ck = 0 , b- k - bk + i 2

c- k + b- k - bk + i 2 + ck + bk - b- k - i 2 +

)

2

(

)

2

+ 4c- k ck = 0 .

(6.9) . (6.10) . . a . , a E + A .

x ( t ) , :
1) = a + i , x ( t ) = e 2) = a - i , x ( t ) = e
ikt

k : - < k - M - 1 ; k : M + 1 k < + ;

ikt

3) = a + b0 + c0 , x ( t ) = C = co n s t ; 4) =

1 2

2a + b- k + bk +

(

b- k - bk + i 2

)

2

+ 4c- k ck ,

x ( t ) = 2c- k e


-ikt

- b- k - bk + i 2 -

(

b- k - bk + i 2

)

2

+ 4c- k ck e

ikt

c- k + b- k - bk + i 2 -

(

b- k - bk + i 2

)

2

+ 4c- k ck 0 ,
-ikt

x ( t ) = bk - b- k - i 2 -

(

2 b- k - bk + i 2 + 4c- k ck e

)

- 2ck e

ikt

260


:

3.

§6



c- k + b- k - bk + i 2 - ck + bk - b- k - i 2 -

( ( (

b- k - bk + i 2

)

2

+ 4c- k ck = 0 , + 4c- k ck 0 ,

b- k - bk + i 2

)

2

x ( t ) = C1e


- i kt

+ C2eikt , C1 C2 ­ b- k - bk + i 2

c- k + b- k - bk + i 2 - ck + bk - b- k - i 2 -
5) =

)

2

+ 4c- k ck = 0 , + 4c- k ck = 0 ;

(

b- k - bk + i 2

)

2

1 2

2a + b- k + bk -

(

2 b- k - bk + i 2 + 4c- k ck ,

)

x ( t ) = 2c- k e


- ikt

- b- k - bk + i 2 +

(

2 b- k - bk + i 2 + 4c- k ck e

)

ikt

c- k + b- k - bk + i 2 +
x ( t ) = bk - b- k - i 2 +

(

b- k - bk + i 2 + 4c- k ck 0 ,

)

2

(

b- k - bk + i 2

)

2

+ 4c- k ck e

- ikt

- 2ck e

ikt



c- k + b- k - bk + i 2 + ck + bk - b- k - i 2 + x ( t ) = C1e
- i kt

(

b- k - bk + i 2 + 4c- k ck = 0 , b- k - bk + i 2

)

2

(

)

2

+ 4c- k ck 0 ,

+ C2eikt , C1 C2 ­

261


:

3.

§7



c- k + b- k - bk + i 2 + ck + bk - b- k - i 2 +
6.2 .

( (

b- k - bk + i 2 + 4c- k ck = 0 ,

)

2

b- k - bk + i 2

)

2

+ 4c- k ck = 0 .

7. (1.1) (1.2) H L . H L H


L2 . 6.1,

(6.1) (6.2). (1.2). a = 0 , (1.1).

a + i + b- Bk = ck


k

, a - i + bk c-
k

k = 1, . . ., M

k = d et Bk = a + i + b-

()(

k

)(

a - i + bk - c- k ck

)

k = 1, . . . , M .

, a , (1.2),

ai



a -i .

(7.1)

6.2, (1.2) H L , 0 a . : 262


:

3.

§7

7.1. (1.2) f ( t ) H L

a + b0 + c0 0 , k 0, k = 1, . . . , M .

( 7 .2 ) (7.3)

7.1 § 8. 0 a , (7.2) (7.3) . (1.2) f ( t ) H L , ­ x ( t ) H L . 7.2. (7.3)

a + b0 + c0 = 0 .

(7.4)

(1.2) f ( t ) H L


1 2

-

f (t )

dt = 0 ,

(7.5)

, (1.2) , d 0 x ( t ) H L ,


1 2

-

x (t )

d t = d0 .

(7.6)

7.2 § 8. , k {1, . . ., M

}



k = 0,

k {1, . . . , M } .
263

(7.7)


:

3.

§7

(1.2) k , (7.7), Bk . 10 :

(A )
k, 1

a + i + b- k = 0, a + i + b- k 0, a + i + b- k = 0, a + i + b- k = 0, a + i + b- k = 0, a + i + b- k 0, a + i + b- k = 0, a + i + b- k 0, a + i + b- k = 0, a + i + b- k 0,

c- k = 0, c- k = 0, c- k 0, c- k = 0, c- k = 0, c- k 0, c- k = 0, c- k = 0, c- k 0, c- k 0,

ck = 0, ck = 0, ck = 0, ck 0, ck = 0, ck = 0, ck 0, ck 0, ck = 0, ck 0,

a - i + bk = 0 ; a - i + bk = 0 ; a - i + bk = 0 ; a - i + bk = 0 ; a - i + bk 0 ; a - i + bk = 0 ; a - i + bk 0 ; a - i + bk = 0 ; a - i + bk 0 ; a - i + bk 0 .

(A )
k, 2

(A )
k, 3

(A )
k, 4

(A )
k, 5

(A )
k, 6

(A )
k, 7

(A )
k, 8

(A )
k, 9

(

Ak

, 10

)

k k = 0 ,

(A )
k, j

, j = 1, . . . ,1 0 .

, :

264


:

3.
- ikt

§7

()
Bk
,1

1 2 1 2 1 2 1 2 1 2 1 2 1 2 ck



-



f ( t ) ei kt d t = 0,
- ikt

1 2



-

f (t )

e

dt = 0 ;

()
Bk
,2



-

f (t )

e

dt = 0 ;

()
Bk
,3



-

f (t )


e

- ikt

dt = 0 ;

()
Bk
,4

-

f (t )

ei kt d t = 0 ;

()
Bk
,5



-

f (t ) f (t ) f (t )

ei k t d t = 0 ;

()
Bk
,6



e

- ikt

dt = 0 ;

-

()
Bk
,7



ei kt d t = 0 ; a + i + b- 2

k

-

(B )
k, 8



2

-



f ( t ) ei k t d t -


-

f (t )

e

- ikt

dt = 0 ;

(B )
k, 9

a - i + bk 2


-



f ( t ) ei k t d t -

c-


k

2

-

f (t )


e

- ikt

dt = 0 ;

(


Bk

, 10

)

2

ck

-



f ( t ) ei kt d t -

a + i + b- 2

k

-

f (t )

e

-ikt

dt = 0

265


:

3.
- i kt

§7

(
(B )
k, j

Bk

, 10

)

a - i + bk 2



-



f ( t ) ei k t d t -

c-


k

2

-

f (t )

e

dt = 0 .

(1.2) ­ , j = 1, . . . ,1 0 , .

, . Ak x ( t

()
,j

, j = 1, . . . ,1 0
- k

)

(1.2). d

d

+ k

­ -

. :

()
C
k, 1

1 2 1 2 1 2 1 2 1 2 1 2 1 2



-



- x ( t ) ei k t d t = d k ,

1 2



-

x (t )

e

- ikt

+ dt = dk ;

()
C
k, 2



-

x (t )

e

- ikt

+ dt = dk ;

()
C
k, 3



-

x (t )


- ei k t d t = d k ;

()
C
k, 4

-

x (t )

e

- ikt

+ dt = dk ;

()
C
k, 5



-

x (t )

- ei kt d t = d k ;

(C )
k, 6



-

x ( t ) ei k t d t = d x ( t ) ei k t d t = d

- k



1 2 1 2



-

x (t ) x (t )

e

- ikt

+ dt = dk ;

(C )
k, 7



-



- k





e

- i kt

-

+ dt = d k ;

266


:

3.

§7

()
C
k, 8

1 2 1 2 1 2



-

x (t ) x (t )

-

e

- i kt

+ dt = d k ;

(C )
k, 9



-

- ei kt d t = d k ;

(

C

k , 10

)



x ( t ) ei k t d t = d

- k



1 2



-

x (t )

e

- i kt

+ dt = d k .

: 7.3. (7.2)

k {1, . . ., M

}

­ (7.7). (1.2)

f ( t ) H L Bk

()
,j

(A )
k, j

,

; , (1.2) ,
- k

d

d

+ k



x ( t ) H L , C 7.3 § 8.

()
k, j

, j {1, . . .,1 0} .

7.4. (7.4)

k {1, . . ., M

}

­ (7.7). (1.2)

f ( t ) H L Bk

()
,j

(A )
k, j

,

(7.5); , (1.2) - k

, d 0 , d

d

+ k



x ( t ) H L ,

(C )
k, j

,

j {1, . . .,1 0} (7.6).
7.4 § 8. 267


: 8. 7.1 ­ 7.4

3.

§8

7.1. 6.1 (1.2) ( ) (6.2). (1.2) (6.2). . (1.2) . , -, (6.2), (7.2). -, k {1, . . ., M

}



, (6.2), . ­ (7.3). . . (7.2) (7.3). (7.2) , x0 (6.2) . (7.3) , (6.2), xk , k = ± 1, . . . , ± M . (7.1) , (6.2) . , (1.2) . . 7.1 . , 7.1, . , . 7.2. (7.1) (7.2). (6.2) :

xk , k 0 .

268


:

3.

§8

(7.4) , x0 , 0 x0 = f 0 . f 0 = 0 , (7.5). , (7.3) (7.4) (6.2) , x0 , . x0 , , . (1.2) f ( t ) H L

d 0 . d 0 = x0
(6.2). , , x ( t ) (1.2), (7.6). 7.2 . 7.3 7.4.

k {1, . . ., M

}

(7.7).

(6.2)

a+i+b x +c x = f , -k -k -k k -k ck x- k + a - i + bk xk = f k .

(

(

)

)

( 8 .1 )

(8.1) , f

-k

f

k

. (8.1) Ak

() (A ),
,j k, 1

, j = 1, . . . ,1 0 . (8.1)

0 x- k + 0 xk = f - k , 0 x- k + 0 xk = f k .

(8.2)

269


:

3.

§8

(8.2) f Bk

()
,1

-k

= 0 f k = 0 , -

.

Bk , x- , x- k = d
- k
k

()
,1

, (8.2) -

xk , ,
+

. xk = d k , -

(6.2). , , , x ( t ) (1.2), C

()
k, 1

. , (8.1)

Ak

()
,2

a + i + b x + 0x = f , -k -k k -k 0 x- k + 0 xk = f k .
a + i + b- k 0 , x-
k

(

)

(8.3)


-k

(8.3) f

.

(8.3) . f k = 0 , Bk Bk

()
,2

()
,2

.

, xk ,
+

, . , xk = d k , (6.2). , ,

270


: , x ( t

3.

§8

)



(1.2), C

() (A ),
k, 2 k, 6

. (8.1)

a+i+b x +c x = f , -k -k -k k -k 0 x- k + 0 xk = f k .

(

)

( 8 .4 )

a + i + b- k 0 c- k 0 , (8.4) f k = 0 ,

(B )
k, 6 k, 6

. ,
-k

, (8.4) f

.

(B )

, x-

k

xk , , .
, x- k = d
- k

xk = d k ,

+

(8.4). , , ,

x ( t ) (1.2),
C

()
k, 6

.

. (7.2) 7.1, (7.4) ­ 7.2. 7.3 7.4 .

271


:

3.

§9

9. (1.1) (1.2). , , (1.1), (1.2) a = 0 , (1.1) (1.2). (1.1) (1.2) , , . (1.1) (1.2)

AN xN = f N ,
(a (a (a AN ) xN ) = f N ) ,

(9.1) ( 9 .2 )



AN : N N ,
(a AN ) :
2N

( 9 .3 ) (9.4)



2N

­ N 2 N ;

fN N , xN N ,

(a fN ) (a xN )

2N

,

(9.5) (9.6)

2N

, . 9.1. . N 2 ­ .

hN =


2 . N

(9.7)

s0
.

,N

-hN , - 0, 5hN

(9.8)

272


:

3.

§9

N

{

s; N

}



{t; N }

, [ 0, 2 ] :

sm t


,N

= s0 =t

,N

+ mhN , + m hN ,

m = 1, . . . , N , m = 1, . . ., N ,

(9.9) (9.10)

m, N

0, N

t
s0
,N

0, N

= s0

,N

+ 0, 5hN .

(9.11)

t0

,N

, (9.8) (9.11), , hN ­ . . . -

{s; N

} {t; N }

[55 .265], (9.9) (9.10) [ 0, 2 ] . {t ; N
{
s;N

}



{

t ;N

}

, {s; N

}

}

x ( t ) H




2, N



{

s;N

}

x = x s1,

( ( ), x (s )
N

, . .., x s

(

N,N

))

T

,

(9.12)



{

t ;N

}

x = x t1,

( ( ) , x (t )
N 2, N

, . .., x t

( ))
N,N

T

.

(9.13)


() m- N = t1, N - s ,
m+1, N

, ,

m = 0,1, . . . , N - 1 , m = 0,1, . . ., N - 1 .

(9.14) (9.15)

() m+ N = t1, N + s ,

m +1, N

AN (1.3)

AN = TN + K1, N + K
273

2, N

,

(9.16)


: TN , K1,
N

3.

§9

K

2, N

­ (1.4), (1.6) (1.7):

TN ­
( -) 1 ct g 0, N N 2

1(, N) 1 ct g N 2
-

. ..

( N -)1, 1 ct g N 2
-

N

;

(9.17)

K1,

N

­

1 ( -) N k1 0, N K
2, N

() ()
(t )

1 - k1 1(, N) N

() ()

. ..

1 (- k1 N -)1, N

(

N

)

;

(9.18)

­

1 (+) N k2 0, N
k1 ( t ) k
2

1 + k2 1(, N) N

.. .

1 (+ k2 N -)1, N

(

N

)

;

(9.19)

(1.8).

x ( t ) H . (T x ) ( t ) H . (T x ) ( t ) {t , N

}

{(Tx ) (t )
1, N

,

(T x )

(t )
2, N

, ... ,

(T x )
{ }

(t )}
N,N

.

(9.20)

(9.20) (T x ) ( t ) H




t ;N

:



{

t ;N

}

(T x )

= ( T x ) t1,

(()
N

, (T x ) t

()
2, N

, . .., (T x ) t

( ))
N,N

T

.


(9.21)

x ( t ) H {s, N

}

{x ( s ) , x ( s )
1, N 2, N

, . .. , x s

(

N,N

)}

.

(9.22)

274


:

3.

§9

(9.22) x ( t ) H



{ }

{

s;N

}

:



s;N

x = x s1,

( ( ), x (s )
N 2, N

, . . ., x s

(

N,N

))

T

.

(9.23)

TN (9.21). [14, . 35]

( (




{

t;N

}

(T x )


{

{

t ;N

}

(T x )

) - (T ) (T
m m

N



{ {

s; N

} }

x x

N


}

s;N

) )

=O N
m

(

-

ln N , m = 1, . . ., N , (9.24)

)

m - m

t ;N

}

(T x )

TN

{

s;N

x.

, ,

( (



{

t;N

}

( (

K1 x

)

) -(
m m

K1, N K

{

s;N

}

x

)

=O N
m

(

-

)

,

m = 1, . . . , N , (9.25)

{

t;N

}

K2 x

)

) -(
m

2, N



{

s;N

}

x

)

=O N
m

(

-

)

,

m = 1, . . . , N . (9.26)

(9.24) ­ (9.26)

(




{

t;N

}

(

Ax

)

) -(
{
t ;N

AN

{

s;N

}

x

)

=O N
m

(

-

ln N , m = 1, . . ., N ,
-

)



}

A - AN

{

s; N

}

=O N
L
2, N

(

ln N ,

)

(9.27)

A

L

=
2, N

s up
y
L 2, N

=1

Ay

L

(9.28)
2, N

275


:

3.

§9

y

L

=
2, N

1 N

m =1



N

y

2 m

(9.29)

­ L2 ; y = y1 , y2 , . . ., y

(

N

)

T

.

9.2. . TN , K1,
N

K

2, N

, § 1 -

(9.17) ­ (9.19), . , - , §§ 1 ­ 3 1 , [21, 23, 75]. TN , K1,
N

K

2, N

,


FN TN FN ,
FN K1, N FN



FN K

2, N

FN ,

FN ­ N (1.1.19). 9.1. N 2
FN TN FN = d i a g 0 , 1 , ... ,

(

N -1

)

,

(9.30)



0 = 0;

k = e

k i - 2 N

, k = 1, . . . , N - 1 .

(9.31)

9.1.

am =

t1, N - s 1 ctg N 2

m +1, N

, m = 0, . . . , N - 1 ,
N -1

TN a0 a1 . . . a (1.29, 1.7, 1),
FN TN FN = d i a g

.

( (w )
N 0

,

N

( w1 )

, ... ,

N

(

wN

-1

))

,

276


:

3.

§9





N

( w) =
N

N -1

m =0

am w m .
: k = 0, . . ., N - 1 . -

sm
,N

( wk )
(1

t

m, N



t1, N - s


m +1, N

=

- 2m ) , m = 0, . . ., N - 1 , N

am =

1 (1 - 2m ) , m = 0, . . ., N - 1 . ctg N 2N

(9.32)

a N = a0 ( (9.32)

m = N ), a1 , a2 , . . . a a

N -1

,a

N

-

,
N +1-m

= - am , m = 1, . . . , N .

(9.33)

, , N . N = 2 N1 ­ .. (9.33),



N

(

w) =

m =1

am (
(

N1

wm - w

N -m +1

)

.

(9.34)


m wk

-

N wk - m+1

2m - 1) k = 2 si n e N

k -i + 2 N

,

k = 0, . . ., N - 1 m = 1, . . ., N1 ,



N

()

wk = -

1 e N1

k - i + N1 2 N

m =1



ct g

(

2m - 1) ( 2m - 1) k . (9.35) sin 2N N

277


:

3.

§9

b(k ) =

N1 -1 m =0



ct g ( 2m + 1) N s i n k ( 2m + 1) N ,
N1 -1 m =0

(9.36)

c(k ) =



co s 2k N + m

(

N

)

,

(9.37)

N =



2N

,

N =


N

.

(9.38)

b ( k + 1) - b ( k ) = =
N1 -1 m= 0



s i n ( k + 1) ( 2m + 1) N - s i n k ( 2m + 1) N c t g ( 2m + 1) N =
s i n N + m

=2

N1 -1 m =0


=

(

N

)

c o s ( 2k + 1) N + m

(

N

)

co s N + m si n
N

( (

N N

+ m

) )

=

N1 -1 m= 0



co s 2 ( k + 1) N + m

(

N

)+ m =0

N1 -1

c o s 2k N + m

(

N

)

=
(9.39)

= c ( k + 1) + c ( k ) .
c ( k )

co s + c o s ( + ) + . . . + co s ( + ( n - 1) ) =
co = n c n -1 n sin s + 2 2 , sin 2 os ,



sin


2

0,
(9.40)



si n


2

= 0,

278


:

3.

§9

, k . (9.37) c ( 0 ) = N1 , (9.40) k = 1, . . ., N - 1 ,

N -1 2kN1 co s 2k N + 1 2k N s i n 2 2 c(k ) = 2 k N sin 2

N

=

s i n k N 2 s i n k

N N

,

(9.41)

N (9.38). k = 1, . . ., N - 1

k N N = k ,


s i n k N N = 0;

k < , N s i n k N 0, k = 1, . . ., N - 1 , 0 < k N =

, , (9.41)

c (1) = c ( 2 ) = . . . = c ( N - 1) = 0 .
(9.39) (9.42)

(9.42)

b (1) = b ( 2 ) = . . . = b ( N - 1) .
b (1) . (9.36)

(9.43)

b (1) =
=
N1 -1 m= 0

N1 -1 m= 0



ct g N + m

(

N

)

s i n 2 N + m

(

N

)

=



1 + c o s 2 + m N

(

N

)

s i n N1 N =N + = N1 . 1 sin N

(9.44)

(9.43) (9.44)

b ( k ) = N1 ,

k = 1, . . ., N - 1 .

(9.45)

(9.43) (9.45) (9.35),



N

( w0 )

= 0;



N

( wk )

=e

k i - 2 N

, k = 1, . . . , N - 1 .

279


: (9.31) .

3.

§9

N = 2 N1 + 1 ­ . m = N1 + 1 (9.32)

am = a

N1 +1

=-

1 + 2 N1 1 1 ctg = - ctg = 0 . N 2N N 2

(

)

(9.34) N . N . 9.1 . 9.2. N 2M + 1 , M ­ (1.8),
FN K1, N FN = d i a g µ1, 0 , µ1,1 , ... , µ1,

(

N -1

)

,

(9.46)



µ1,

n

b0 , n -i b- n e N , = 0, ( N -n ) b ei N , N -n

n = 0, n = 1, . . . , M , n = M + 1, . . . , M + M 0 , n = M + M 0 + 1, . . . , 2M + M 0 .
(9.47)

9.2. N 2M + 1 . M 0 0 , N = 2 M + 1 + M 0 .

a1, m =

1 - k ( ) , m = 0, . . . , N - 1 , N 1 m, N

(

)

(9.14),

a1, m =

1 (1 - 2m ) k , m = 0, . . ., N - 1 . N 1 N
280

(9.48)


:

3.

§9

(1.8), (9.48),

a1, m =

1 N

k =- M



M

bk e

i

k (1- 2 m ) N

, m = 0, . . ., N - 1 .

(9.49)

(1.29, 1.7, 1),
FN K1, N FN = d i a g 1,

( (w )
N 0 N

, 1,
N -1

N

( w1 )

, ... , 1,

N

(

wN

-1

))

,



1,

( w) =
N

m =0

a1, m wm .

(9.50)

1,

( wn )

: n = 0, . . . , N - 1 . (9.49) (9.50)

1,

N

( wn ) = m =0 k =- M
1 N
M

N -1

bk e

i

k (1- 2 m ) N

m wn

1 = N

k =- M

bk

M

N -1 i k (1- 2 m ) - i 2 mn eNeN

=

m =0

1 = N

k =- M



M

bk e

i

2( k + n ) k N -1 -i m N N e m =0



= µ1, n ,

µ1, n ­ (9.47). (9.46) .
FN K1, N FN

b-1 -) K1(,1 =

b- 2

2

b-
(
M -1)



M -1

b-

M

, M

(9.51)

281


:

3.

§9

=e

i


N

;

(9.52)

(- K 2,2) ­ M 0 ( M 0 = 0 -

K

( -) 2,2

);

b M ( -) K 3,3 =

M

bM -1

M -1

b2
2

; b1

(9.53)

( -) ( -) K1,1 K 3,3 .

FN

K1,

N

FN =

b0 0 0 0

0
- K1(,1 )

0 0
(- K 2,2)

0 0

0

0 0 . 0 ( -) K 3,3
-

9.2 . (9.7) (9.8). 9.3. N 2M + 1 , M ­ (1.8),
FN K 2, N

N

= 2s0

,N

+ 2, 5hN , hN s0

,N

FN = p d µ

(

2, 0

, µ2,1 , ... , µ

2 , N -1

)

,

(9.54)

282


:

3.

§9

µ

2, n

=

c0 , c- n e 0, c
N -n - i n
n

n = 0, , n = 1, . . . , M , n = M + 1, . . . , M + M 0 , e
- i( N - n )
n

(9.55)

, n = M + M 0 + 1, . . ., 2 M + M 0 .

9.3. 9.2,

M 0 0 , N = 2 M + 1 + M 0 .
(9.9), (9.10) (9.15)
() m+ N = 2 s0 ,
,N

+ ( m + 2, 5 ) hN = N + m hN , m = 0, . . ., N - 1 , (9.56)
= 1 (+) k2 m, N , m = 0, . . ., N - 1 , N



a

2, m

(

)

(9.56),

a

2, m

=

1 k + mhN , m = 0, . . . , N - 1 . N2 N

(

)

(9.57)

(1.8), (9.57),

a

2, m

=

1 N

k =- M



M

ck e

i k N + m hN

(

) , m = 0, . . . , N - 1 .

(9.58)

(2.12, 2.5, 1),
FN K 2, N

FN = p d w0

(

2, N

( w0 )

, w1
N -1

2, N

( w1 )

, ... , wN -1

2, N

(

wN

-1

))

,





2, N

( w) =
2, N

a

m =0

2 , N -1- m

wm .

(9.59)



( wn )

: n = 0, . . . , N - 1 . (9.58) (9.59)

283


:

3.
- i mn hN

§9



2, N

1 ( wn ) = N m =0 k =- M
M

N -1

ck e

i k N - hN - mhN

(

)e

=

=
µ

1 N

k =- M



M

ck e

i k

N

e

- i( n + k )

N -1
N

m= 0



e

- i ( k + n )hN m

=µ

2, n

,

2, n

­ (9.55).

(9.54) .
FN K 2, N

FN

(+) K1,3 = c -M


c- 2 c-
M M +1

2



M -1

c-1 ,

(9.60)

=e

i

N

;

(9.61)

(+ K 2,2) ­ M 0 ( M 0 = 0 -

K

(+) 2,2

);

(+) K3,1 = c2 c 1

c

2

M -1



M -1

cM M ;

(9.62)

(+) (+) K1,3 K 3,1 .

284


:

3.

§9

FN

K

2, N

FN =

c0 0 0 0

0 0 0
(+ K 3,1)

0 0
(+ K 2,2)

0

0 + K1(,3 ) . 0 0

9.3 . 9.1 ­ 9.3 : 9.1. N 2M + 1 , M ­ (1.8),

b0 + c0 0 FN AN FN = 0 0

0
- 1,1 + K1(,1 )

0 0
2,2

0
(+ K 3,1)

0

+ K1(,3 ) , 0 (- 3,3 + K 3,3) 0
2

(9.63)

b0 , c0 ­ k1 ( t ) k

(t )

( -) ( -) (1.18), K1,1 , K 3,3 ,

+ (+ K1(,3 ) K 3,1) ­ (9.51), (9.53), (9.60) (9.62), -

,

1,1 = d i ag 1 , ... , M , 2,2 = d i ag M +1 , ... , M + M , 0 = d i ag , ... , M + M M + M 0 +1 3,3

(

( (

)

)

;
+M

0

)

,

m , m = 1, . . . , N - 1 ­ (9.31); N = 2 M + M 0 + 1 .

285


:

3.

§ 10

10. , (1.1) 7.1 ­ 7.4). ­ (9.1): ( -

AN xN = f N ,

(10.1)

AN ­ (9.3), (9.16); (9.5)

fN =
­ ,

{

t; N

}

f = f t1,

(( )
N

, ... , f t1,

( ))
N

T

xN =
­ .
yN = FN xN ,

{

s; N

}

x

g N = FN f N .

(10.2)

(9.16) AN , : FN = FN (10.2), (10.1)
-1

(

FN TN FN + FN K1, N FN + FN K

2, N

FN y N = g N .

)

(10.3)

(10.3)
FN TN FN + FN K1, N FN + FN K

2, N

b0 + c0 FN = 0

0 , DN

(10.4)

286


:

3.

§ 10

( -) 1,1 + K1,1 DN = 0 (+) K 3,1

0
2, 2

+ K1(, 3)

0



3, 3

0 . (- + K 3, 3)

DN

M , M0, M

(

N = 2 M + M 0 + 1 .

)

(10.4) . , M 0 + 1 M :

(
k ( y

b0 + c0

) ( y N )0 = ( g N )

0

,

(

k = 0) ,
0

(10.5)

Nk

) = ( g N )k , (

k = M + 1, . . ., M + M

)

,

(10.6)

+b k y + c- k k yN Nk k -k k k yN ck yN k + N - k + bk ( k = 1, . .., M ) ,

(

)( ) ()(

( ) N - k = ( g N )k , ) ( ) N -k = ( g N ) N -

k

,

(10.7)

(9.52) (9.61), y k - y
N

( N )k ( g N )

k

­ -

gN .

10.1. b0 + c0 0 , k 0 . (10.1). k 0 , k = 1, . . ., M , 0 > 0 ,

k 0 ,

k = 1, . . ., M .
287

(10.8)


:

3.

§ 10

, N = 2 M + M 0 + 1 . M , (1.8), N ­

{

s; N } { t; N } . N
0, N

M 0 M N

. , 0 > 0

( 0)

, k = 1, . . ., M

2 k 2 k bk - 1 - e x p i b < 0 , 1 - e x p -i N N -k
( 0, 1) ­ .

(10.9)

10.1. 7.1 (10.9). f H xN = AN f
-1
N



(10.1) -


{ }

- AN1 f N -

s; N

A- 1 f
L2
,N

=O N

(

-

ln N .

)

(10.10)

10.1.
- AN1 f N -

{

s; N

}

- A-1 f = AN1

{

t; N

}

f -
}

{

s; N

}

A- 1 f =
(10.11)

- = AN1

(

{

t; N

}

A - AN

{

s; N

)

A- 1 f .

(10.11),
- AN1 f N -

{

s; N

}

A- 1 f
L2
,N

A-1 f
L2
,N

- AN

1
L2
,N



{

t; N

}

A - AN

{

s; N

}

L2

.
,N

(10.12)

288


: 0 (10.8),
- AN1

3.

§ 10

L2

,N

1 b ma x 1, , , b0 + c0 0
,N L 2

(10.13)



b = max Bk

,N

: k = 1, . . . , M .

(10.12), (10.13) (9.27) (10.10). 10.1 . (10.1), t . , [105, 107] FN , FN = FN xN , O N lo g 2 N
-1

(

)

-

. N ­ FN xN . , TN , K1,
N

K

2, N

(9.17) ­ (9.19) :

qn =

() 0, N 1 ct g N 2
-

( N -)1, 1 ct g N 2
-

N

k 1 , q = 1, n k 1

( ( ) )
- 0, N

N
( N--)1,
N

(

)

N

k , q2, n = k 2


2

( ( ) )
+ 0, N

N

(

( N+-)1,

N

)

N

.

1. FN TN FN , FN K1, N FN FN K 2, N

FN ; FN qN , FN q1,

N

FN q2


,N

;

, (9.30), (9.47) (9.55); FN TN FN , 289


:
FN K1, N FN FN K

3.

§ 10

2, N

FN (10.3) -

2 N ( ). 2. g
N = FN f N ;

3. (10.5) M 0 (10.6);

( yN )

0



( yN )

k

, k = M + 1, . . ., M + M 0 ;

M (10.7), ;
y

( N)

k

, k = 1, . . ., M

k = M + M 0 + 1, . . . , 2 M + M 0 .
4. xN = FN y N . , , , 1 2 3 4 :

O N lo g 2 N

( ( (

O( N O( N

) )

) ) )

O N lo g 2 N O N lo g 2 N

10.2. b0 + c0 = 0 , k 0 .

b0 + c0 = 0,

k 0, k = 1, . . ., M .

7.2 a = 0 , (1.1) -

290


:

3.

§ 10

(7.5), d 0 (1.1),

1 2



-

x (t )

d t = d0 .

, d 0 = 0 . , (1.1) (10.5):


(

Ax ) ( t ) = f ( t ) ,

1 2

-

x (t )


dt = d

0

(10.14)



(

Ay ) ( t ) = f ( t ) ,

1 2

-

y (t )

dt = 0 .

(10.15)

(10.14) (10.15)

x ( t ) y ( t ) + d0 ,
, d 0 = 0 .

t [ - , ] .

(

Ax ) ( t ) = f ( t ) ,

1 2



-

x (t )

dt = 0



( Ax ) ( t )


= (t ) , f

(10.16)

()

( Ax ) ( t ) x (t ) = 1 A ; x ( t ) dt 2 -

A: H H â ,

(10.17)

( t ) = f (t ) . f 0
291

(10.18)


: H 0 H


3.

§ 10

­

0

; , A H
H 0 â {0} .



(9.12) (9.13)



{

s; N

x = { s, N } 0

x }

;



{

s; N

}

: H

N +1

,

(10.19)



{

t; N

x { t ; N = } c c

x }

;



{

t; N

}

: H â

N +1

.

(10.20)

(10.16)

AN xN = N , f


(10.21)

AN AN = T aN

0 , 0

1 T aN = . .. N
, = fN

1 N

N ,

(10.22)

xN xN = xN , N
x
N,N

{

t; N

}

, f

(10.23)

­ , . .

[52] . , (10.17) (10.22)

A AN
AN
x ( t ) H 0 .


{

s; N

}

x-

{

t; N

}

Ax
L2
, N +1

=O N

(

-

ln N

)

(10.24)

292


: ,

3.

§ 10

AN

{

s; N

}

x-

{

t; N

AN { s ; N } x - x = A T } aN s; {


N

{

t; N

}

}

x

Ax ,

, ,

AN

{

s; N

}

x-

{

t; N

}

Ax
L2
, N +1

=

=

1 N+


1


k =1

N

(

AN

{

s; N

}

x-

{

t; N

}

Ax

)(
2 k

T + aN

{

s; N

2 x }

)

1 N



N

k =1

(

AN

{

s; N

}

x-

{

t; N

}

Ax

)

2 k

+

1 N


k =1 N

N

x sk

()
,N ,N

=

= AN

{

s; N

}

x-

{

t; N

}

Ax
L2
,N

+

1 N



k =1

x sk

()

0

.

(10.25)

(10.25) (9.27). x ( t ) H ,

(10.25)

1 N



N

xs

k =1

()
k,N

=O N

(

-

)

.

(10.26)

(10.24) . (10.21). (10.21) , , . , (1.1) A T , K1 = 0 , K 2 = 0 [52], , , ­ . , N , . (10.21) 293


:

3.

§ 10

, § 7 1. ,

eN = ( 0, ... , 0, 1)
T T

N +1

,
N +1

(10.27)

1 1 gN = , ... , , 0 N N

(10.28)

AN

AN : AN eN = 0,




AN g N = 0 .

(10.29)

BN = g N eN ,

(10.30)

(10.21)

x CN N = hN ,


(10.31)

CN = AN + BN ,


N f hN = N - g N g N , f

(10.32)

= CN

(N (N a1 1 ) ... a1N )





(N (N aN 1 ) ... aNN)

1 N

...

1 N

1 , N 0

1 N

AN =

(N (N a1 1 ) ... a1N ) . (N (N aN 1 ) ... aNN)



294


:

3.

§ 10

10.2. 7.2 f (10.9). ( t ) H 0 â {0} (10.31)

-1 , xN = C N hN


- C N1hN -

{

s; N

}

f A- 1
L2
, N +1

=O N

(

-

ln N .

)

(10.33)

10.2. (10.32)

AN C

N

7.2. , (10.31) (10.21):

- + f C N1hN = AN N .

(10.34)

(10.31)

FN FN = 0
FN ­ . (10.31)

0 , 1

(10.35)

(


FN C N FN FN xN = FN hN .

)

(10.36)

F A F C F = N N N FN N N T aN FN
T a N FN = FN a

FN a N , 0

(10.37)

(

N

)

T

= (1, 0, . . . , 0 ) N .

(10.38)

295


: (9.63), (10.37) (10.38),

3.

§ 10

0 0 FN C N FN = 0 DN 1 0


1 0 , 0 0 . (- + K 3,3)

(10.39)

( -) 1,1 + K1,1 DN = 0 (+) K 3,1

0
2,2

+ K1(,3 )

0



3,3

(10.39) , (-

x FN N ) (10.36) .
(10.6) (10.7). 10.1. (10.33). (10.13)

b - DN1 max 1, , 0

(10.40)

0 b (10.8) (10.9). (10.39) (10.40)
-1 N - m0 DN1 ,

C

(10.41)

m0 ­ , N . (10.24), (10.26) (10.41) (10.33). 10.2 .

N = FN xN , y
296

N = FN hN . z

(10.42)


:

3.

§ 10

z N ( (10.42), (10.23), (10.20), (10.18)),


z ( N )k = (

Fn f

Nk

)

, k = 1, . . . , N - 1; N z

()

0

= 0, N z

()

N

= 0 . (10.43)

(10.36) :

y z ( N )N = ( N )
+b k -k k y ck N

0

= 0,

k =0,

(10.44)

(

y )( ) ( ) + (
k k

Nk N -k

+ c- k k N y + bk
k

z ( ) N -k = ( N )k , y z ) ( N ) N -k = ( N ) N -

k

,,

(10.45)

k = 1, . . ., M

k ( N y

z )k = ( N )

k

,

k = M + 1, . . ., M + M 0 ,
N

(10.46) (10.47)

y z ( N )0 = ( N )

= 0,

k=N.

(10.36). 1. 1 10.1. 2. 2 10.1. 3. M (10.45), ;

y ( N )

k

, k = 1, . . ., M

k = M + M 0 + 1, . . . , 2 M + M 0 ; M


0

(10.46);

y ( N ) y ( N )0 =

k

, k = M + 1, . . ., M + M 0 ; (10.44) (10.47)

0 N y

()

N

=0.

4. 4 10.1. , , , 10.1.

297


:

3.

§ 10

10.3. b0 + c0 0 , k = 0 . b0 + c0 0 k = 1

i + b- Bk = ck

k

, -i + bk c-
k

§ 7, , k :

b0 + c0 0;

d et B1 = 0;

d e t Bk 0, k = 2, . . . , M .

§ 7, k ( ,

k = 1 ) 10

(A )-(A )
1,1 1,1 0 0

a = 0 . -

(1.1) ,

(B ) - (B )
1,1 1,1 0

a = 0 . C1,1 - C1,1

( )( )

-

(1.1). , (1.1) 10 , . , A1,

()
2

(A )-(A )
1,1 1,1 0

, -

.

7.3

(
1 2

Ax ) ( t ) = f ( t ) , e -i s d s = d1+ ,

(10.48)



-

x(s)

(10.49)

d 0 ,

1 2



-

f (t )

e -it d t = 0 .

298


:
+

3.

§ 10

, d1 = 0 . , (10.48)

1 2
+ it



-

x(s)

e -i s d s = 0

(10.50)

d1 e , (10.48), (10.49) d1 . , (10.48), (10.50):
+

(

Ax ) ( t ) = f ( t ) ,

1 2



-

x(s)

e -i s d s = 0 .

(10.51)

(10.51)

( Ax ) ( t )


= (t ) , f

(10.52)

()

x (t ) = 1 A 2

(

-

Ax ) (t



; -is x ( s ) e ds

)

A: H H â ,

(10.53)

( t ) = f (t ) H â . f 0
H1 H


­ x ( t ) , -it

, x ( t ) e

.

7.3 A H


1

H 0 â {0}

. (10.52)

x AN N = N , f

(10.54)

299


:

3.

§ 10

AN AN = bN

0 , 0

1 bN = e N
xN xN = xN , N ,
s; N

- i s1,

N

.. .

1 e N

-i s

N,N

N , (10.55)

= fN
}

{

t; N

}

. f

(10.56)

A , AN ,

{



{

t; N

}

, (10.53),

(10.55), (10.19) (10.20) ,

AN
x ( t ) H1 .


{

s; N

}

x-

{

t; N

}

Ax
L2
, N +1

=O N

(

-

ln N

)

,

AN

{

s; N

}

x-

{

t; N

AN { s ; N } x - Ax = T } bN s; {


N

{

t; N

}

}

x

Ax ,

, ,

AN

{

s; N

}

x-

{

t; N

}

Ax
L2
, N +1

=

=

1 N+
1 N
N

1


k =1

N

(

AN

{

s; N

}

x-

{

t; N

}
2 k

Ax

) +(
2 k

T bN

{

s; N

2 x }

)





k =1

(

AN

{

s; N

}

x-

{

t; N

}

Ax

)

+

1 N


N

N

e

-is

k,N

xs

k =1 -is

()
k, N ,N

=

= AN

{

s; N

}

x-

{

t; N

}

Ax
L2
,N

+

1 N



e

k,N

k =1

x sk

()

.

(10.57)

300


:

3.

§ 10

(10.57) (9.27). x ( t ) e
-it

,

(10.57),

1 N
= 1 N


,N

N

e

- i sk

,N

k =1

x sk


()
,N

=



N

e

- i sk

,N

k =1

x sk

()

-

1 2

-

x(s)

e -is d s = O N

(

-

)

.

(10.58)

eN (10.27) AN . ,

b gN = N 0

N +1

,

N -1 / 2ei s1, bN = is N - 1 / 2e N

N

,N



N

AN : AN eN = 0, AN g N = 0 .
(10.59)



.

BN = g N eN .
(10.54)

x CN N = hN ,


(10.60)

CN = AN + BN ,


N f hN = N - g N g N , f

(10.61)

301


:

3.
i s1

§ 10

(N) a1 1 = CN N a( 1 ) N 1 -i e N

...

(N a1N )

...
s1
,N

1 e N 1 e N
N,N

,N

N a( N) N

is

N,N

...

1 e N

-i s

0

.

. 10.3. 7.3 (10.9). ( t ) H 0 â {0} (10.31) f

, x


N

- = C N1hN
-

- C N1hN - {

s; N

}

A- 1 f
L2
, N +1

=O N

(

ln N .

)

(10.62)

10.3.

(10.61) hN R AN

()

C

-1 N



7.3. , (10.60) (10.54):

- + f CN1hN = AN N .
(10.60) (10.35). (10.60)

(

x FN CN FN FN N = FN hN .
FN bN , 0

)

(10.63)



F A F C F = N N N FN N N bN FN
302

(10.64)


:

3.

§ 10

FN bN =

0 , b F = 0 ... 0 e NN 0 i s1, N e

- i s1

,N

.

(10.65)

(9.63), (10.64) (10.65),

DN FN C N FN = 0 0

0 0 e
- i s1
,N

0 is e 1, N , 0

(10.66)

b0 + c0 0 DN = 0 0

0
- 1,1 + K1(,1 )

0 0
2 ,2

0 K ( + ) 1,3 0
3,3

0 K ( + ) 3,1

0

(- + K 3,3)

,

(10.67)

:

K ( + ) ­ K ( + ) , 1,3 1,3 K ( + ) ­ K ( + ) , 3,1 3,1
3,3

( - + K 3,3) ­

3,3

(- + K 3,3) -

, . (10.39) , -

x ( FN N ) (10.36) . (10.6) (10.7). 10.1.

303


:

3.

§ 10

(10.62). (10.13) (10.40),

1 b - DN1 max 1, , , b0 + c0 0
(10.66), (10.67) (10.68)

(10.68)

0 b (10.8) (10.9). -

C

-1 N

- m0 DN1 ,

(10.69)

m0 N . (10.69), (10.57), (10.58), (10.62). 10.3 .

z (10.42). N ( (10.42), (10.56), (10.20), (10.18))

z ( N )k = (

Fn f

Nk

)

, k = 0, . . ., N - 1; N z

()
, ,

N -1

= 0, N z

()

N

=0.

, (10.60) :

(

b0 + c0

y z ) ( N )0 = ( N )

0

k =0, k =1,

(10.70) (10.71)

(

1 + b-1
k

y z ) ( N )1 = ( N )1

+b k -k k y ck N

(

y )( ) ( ) + (
k

Nk N -k

+ c- k k N y + bk
k

z ( ) N -k = ( N )k , y z ) ( N ) N -k = ( N ) N -

k

,,

(10.72)

k = 2, . . . , M

k ( N y

z )k = ( N )k , k = y z ( N )N -1 = ( N ) N = 0, y z ( N )N = ( N ) N -1 = 0,
304

M + 1, . . ., M + M 0 , k = N -1 . k=N.

(10.73) (10.74) (10.75)


: (10.63).

3.

§ 10

1. 1 10.1. 2. 2 10.1. 3. M - 1 (10.72), ;
0

y ( N ) y ( N )

k

,

k = 2, . . . , M k = M + M 0 + 1, . . . , 2 M + M 0 - 1 ; M


(10.34);
0

y ( N )

k

, k = M + 1, . . ., M + M 0 ;

,

y ( N )

N -1

N y

()

N

(10.70), (10.75) (10.74).

4. 4 10.1. , , , 10.1.

305


:

3.

§ 11

(10.63). 1. 1 10.1. 2. 2 10.1. 3. M - 1 (10.72), ;
0

y ( N ) y ( N )

k

,

k = 2, . . . , M k = M + M 0 + 1, . . . , 2 M + M 0 - 1 ; M


(10.34);
0

y ( N )

k

, k = M + 1, . . ., M + M 0 ;

,

y ( N )

N -1

N y

()

N

(10.70), (10.75) (10.74).

4. 4 10.1. , , , 10.1.

11. , , (1.2)

a x ( t ) + ( Ax ) ( t ) = f ( t ) ,

f (t ) H



(11.1)

. , § 9 (9.2). {s; N (9.9) (9.10), . (11.1) t tk , N {t ; N } , (1.4), (1.6) (1.7) , {s; N } :

} {t

; N } ,

ax t

()
k, N

+

1 N

m =1



N

q

km

xs

(

m, N

) = f (t )
k, N

,

(11.2)

305


:

3.

§ 11



q

km

= ct g

t

k, N

-s 2

m, N

+ k1 t

(

k, N

- sm

,N

)+k (
2

t

k, N

+ sm

,N

)

,

k1 ( ) k2 (

)

(1.8).

q11 ... q1N AN = q N 1 ... qN N

,

(11.3)

fN = { xN =
{

t; N

} }

f = f t1, x = x s1, x = x t1,
}


(( )
N N

, . . ., f t

(

N,N

))

T

,

s; N

( ( ) , . . ., x ( s ) )
N, N

T

,

x = N

{

t; N

}

( ( ) ( ))
N

T

, .. ., x t
}

N,N

,



{

s; N

{

t; N

(9.12) (9.13).

, (11.3) (9.16). (11.2)

a x + AN xN = f N . N

(11.4)

(11.4) ( ­ (11.2)) N 2 N . N , . (11.1) t sk , N

{

s; N } , (1.4), (1.6) (1.7) ,

{t ; N } :

a x sk

()
,N

+

1 N

m =1



N

qk m x t

( ) = f (s )
m, N k, N

,

(11.5)

306


:

3.

§ 11



qk

m

= ct g

sk

,N

-t 2

m, N

+ k1 s

(

k, N

-t

m, N

)+k (
2

sk

,N

+t

m, N

)

.



q11 ... q1N A = N q N 1 ... qN N

,

fN = {

s; N

}

f = f s1,

(( )
N

, .. ., f s

(

N, N

))

T

.

(11.5)

N AN x + a xN = f N .
, , N 2 N

a x + AN xN = f N , N AN x + a xN = f N . N

(11.6)

. FN , (11.6)

a y + F A F y = g , N NNNN N FN A FN y + a y N = g , N N N

y = FN x , y N = FN xN , g N = FN f N , g = FN f N . N N N

(11.7)

FN AN FN (9.63). -

9.1 ­ 9.3. FN AN FN .

307


:

3.

§ 11

AN ­ TN , K1, N K 2, N .

TN , K1, N K 2, N
s1, N - t1, 1 ct g 2 N
1 N k1 s1, N - t1,
N

s1, N - t 1 ct g 2 N
1 k s -t N 1 1, N

2, N

...

s1, N - t 1 ct g 2 N
1 k s -t N 1 1, N

N,N

,

(

N

) )
(t )

(

2, N

) )

. ..

(

N,N

)

, (11.8) , (11.9)

1 N k2 s1, N + t1,

(

N

1 k s +t N 2 1, N

(

2, N

...

1 k s +t N 1 1, N

(

N,N

)

k1 ( t ) k

2

(1.8).

x ( t ) H . (9.24) ­ (9.26)

( ( ( (




{

s;N

}

(T x )
K1 x

) -(
m m

TN

{

t;N

}

x

)

=O N
m

(

-

ln N , m = 1, . . ., N ,
-

)



{

s;N

}

( (

) )

) -( ) -(
m

K1, N

{

t ;N

}

x

)

=O N
m

(

)

,

m = 1, . . . , N , m = 1, . . . , N .



{

s;N

}

K2 x

K 2, N

{

t ;N

}

x

)

=O N
m

(

-

) )

,


{ }

s;N

(

Ax

)

) -(
m

AN
}

{

t ;N

}

x

)

=O N
m

(

-

ln N , m = 1, . . ., N , (11.10)



{

s;N

A - AN

{

t ;N

}

=O N
L
2, N

(

-

ln N ,

)

(11.11)

308


:

3.

§ 11

FN AN FN FN TN FN , FN K1, N FN FN K 2, N FN . -

( (11.1) ­ (11.3)). 11.1. N 2
FN TN FN = d i a g 0 , 1, ... , N

(

-1

)

,



0 = 0;

k = e

k i + 2 N

, k = 1, . . . , N - 1 .

(11.12)

11.1.

am =

s1, N - t 1 ctg N 2

m +1, N

, m = 0, . . . , N - 1 , am wm .



N

( w) =

N -1

m =0

1.7 ( 1),
FN TN FN = d i a g
-i 2 n N

( (w )
N 0
N

,

N

( w1 )

, ... ,

N

( N -1 ) )

,

wn = e

(

n = 0, . . . , N - 1) ­ N - 1.

,

( w)
,



N

( w)

, -

9.1,



N

( wn )
N

= wn

N

( wn )

(

n = 0, . . ., N - 1) .



( w0 )

N

= 0

( w0 )

= w0

N

( w0 )

=0.

309


:

3.
n i - 2 N

§ 11

n = 1, . . . , N - 1 ,

N

( wn )

=e





N

()

wn = wn

N

()

wn = e

i

2 n i - n N e 2 N

=e

n i + 2 N

.

11.1 . -

TN TN .
TN -

N . TN
, N . i -i -

TN TN .
11.2. N 2M + 1 , M ­ (1.8),
FN K1, N FN = d i a g µ1, 0 , µ1,1 , ... , µ1,

(

N -1

)

,



b0 , n i b- n e N , µ1, n = 0, ( N + n ) b ei N , N -n

n = 0, n = 1, . . ., M , n = M + 1, . . ., M + M 0 , n = M + M 0 + 1, . . . , 2 M + M 0 .
, (11.13)

11.2. N 2M + 1 . M 0 0 , N = 2 M + 1 + M 0 .
310


:

3.

§ 11

a1, m =

1 k s -t N 1 1, N

(

m, N

)

, m = 0, . . . , N - 1 ,
a1, m wm ,



1, N

( w) =

N -1

m =0

1.1.7),
FN K1, N FN = d i a g

( (w )
1, N 0

,

1, N

( w1 )

, ... ,

1, N

(

wN

-1

))

.



1, N

( w)

1,

N

( w)

,

9.2,



1, N

( wn )

= wn1,

N

( wn )

,

(

n = 0, . . . , N - 1) .

(9.47) (11.13). 11.2 .

b-1 ( -) K1,1 =

b-2

2

b-
(
M -1)



M -1

b-
M

M

, M

(11.14)

b M ( -) K 3,3 =


bM -1

M -1

b2
2

, b1

(11.15)

=e

i


N

,

( - ) K 2,2 ­ M 0 .
311


:

3.

§ 11

FN K1, N FN

FN K1, N FN =

b0 0 0 0

0 - K1,(1 ) 0 0

0 0 - K 2(,2 ) 0

0 0 . 0 - K 3(,3 )

-

K 2

,N

.

11.3. N 2M + 1 , M ­ (1.8),
FN K 2, N FN = p d µ2, 0 , µ2,1 , ... , µ 2

(

, N -1

)

,



µ2, n =

c0 , c- n e 0, c
N -n - i n
N

n = 0, , n = 1, . . . , M , n = M + 1, . . . , M + M 0 , e
- i( N - n )
N

, n = M + M 0 + 1, . . . , 2M + M 0 . K
.

11.3

9.3, N K 2

,N

2, N

FN K 2, N FN :

FN K 2, N FN =

c0 0 0 0

0 0 0
(+ K 3,1)
312

0 0
(+ K 2,2)

0

0 + K1(,3 ) . 0 0


:

3.

§ 11

(+) K1,3 = c -M

c-2 c-
M M +1

2



M -1

c-1 ,

(11.16)

(+) K 3,1 = c2 c 1


cM -1
2

M -1

cM M ,

(11.17)

=e

i

N

,

( + ) K 2,2 ­ M 0 .

11.1 ­ 11.3 . 11.1. N 2M + 1 , M ­ (1.8), 0 0 0 b0 + c0 ( -) (+) 0 0 K1,3 1,1 + K1,1 , FN AN FN = 0 2,2 0 0 + - 0 K 3(,1 ) 0 ,3 + K 3(,3 ) 3 b0 , c0 ­ k1 ( t

)

k

2

(t )

( - ) (1.18), K1,1 ,

- + + K 3(,3 ) , K1,(3 ) K 3(,1 ) ­ (11.14), (11.15), (11.16) (11.17), ,

m , m = 1, . . .

1,1 = d i a g 1, ... , M , ,2 = d i ag M +1 , ... , M 2 ,3 = d i ag 3
M + M 0 +1

(

( (

)

+M

0

)

,



, ... , M

+M0 +M

)

,

, N - 1 ­ (11.12); N = 2M + M 0 + 1 .
313


:

3.

§ 12

12. (1.2) (11.6). (11.6) 2 N

2 N .

aEN AN = AN

AN , aEN

AN AN TN , K1, N , TN , K1, N , (9.17), (9.18), (11.8), (11.9), K (9.19), (11.9). E N . FN , (11.6) (11.7). (11.7) . M 0 + 1 ,
N 2, N

, K2

,N

,



N = 2 M + M 0 + 1 .
a y + b + c y N0 0 0 b0 + c0 y 0 + a y N

()( ( )( )
() ()

) ( N )0 = ( g N ) ( N )0 = ( g N )
n n

0 0

,
(12.1)

,

a y + y = gN Nn k Nn k y n + a yN n = g N N n = M + 1, . . ., M + M 0 .

()() ()()

, ,
(12.2)

314


:

3.

§ 12

M



( )n + (b-n + i ) n ( yN )n + c-n n ( yN ) N -n = ( g N a ( y ) + cn n ( y N ) + ( bn - i ) n ( yN ) = ( gN N N -n n N -n ( b-n + i ) n ( yN )n + c- n n ( yN ) N -n + a ( yN )n = ( g N cn n ( y ) + ( bn - i ) n ( y ) + a ( yN ) = ( g Nn N N -n N N -n
a y N n = 1, . . ., M .

)n , )N - )n , )N -

n

,
(12.3)

n

,

:

( - a ) 2 - ( b0 + c

0

)

2

= 0,
(12.4) (12.5)

1 = a + b0 + c0 , 2 = a - b0 - c0 ,
( (12.1));

( - a)2 +1 = 0
1, 2 = a ± i ,

,

( (12.2) n = M + 1, . . ., M + M 0 );

( - a ) 4 - ( b- n + i ) + ( bn - i ) + 2c- n cn ( - a ) 2 +
2 2

+ ( b- n + i ) ( bn - i ) - c- n c n = 0 ,
2

1, 2 = 2a + b- n + bn ±
= 1 2 2a - b- n - bn ±

1 2

( (

b- n - bn + i 2 b- n - bn + i 2

) )

2

+ 4c - n c n , + 4c - n cn ,

(12.6)

2

3, 4

(12.7)

( (12.3) n = 1, . . ., M ).
315


:

3.

§ 12

(6.8), , (12.5) (12.7) . (12.5) (12.7) , (11.5) {s; N

}

{t; N }

t s .

ax ( s ) + ( Ax ) ( s ) = f ( s ) ,

f (s) H

(11.1). , (11.6)

a x ( t ) + ( Ax ) ( t ) = f ( t ) , a x ( s ) + ( A x ) ( s ) = f ( s ).

(12.8)

(12.3), (6.8), (12.8). , , , (11.6) (12.5) (12.7). , - (12.8). . (12.1) ­ (12.3) N N 2M + 1 ,

M ­ (1.8). , N 0 2M + 1 A N ,
0

N 2M + 1 A

N


-1 N0

A

-1 N

=A



N 2M + 1 .

(12.4) ­ (12.7) ,

C > 0 , , N 2M + 1

A

-1 N

=C



N 2M + 1 .

(12.9)

316


:

3.

§ 12

:

f (t ) F (t, s ) = , f ( s ) x (t ) X (t, s ) = , x ( s )

{t ; N } f , FN {ts;; N } F = { N} {s ; N } f {t ; N } x , {ts;; N} X = { N} {s ; N } x

X

N

ax ( t ) + ( Ax ) ( t ) AX = . a x ( s ) + ( Ax ) ( s )
(12.8) (11.6) :

AX = F , AN X
N

(12.10) (12.11)

= FN .

12.1. (12.10) f H L , (12.4) ­ (12.7) . 12.1 7.1. x H


f H



­ . (9.27),

(11.10) (11.11)

(
Y

{ts;; N } ( A X { N}

) - AN
m

)(

{s ; N } {t ; N }

X

)

m

=O N

(

-

ln N ,

)

N} {ts;; N } A - A N {ts;; N } { N} {

L

=O N
2, 2 N

(

-

ln N ,

)

(12.12)

( N)

m

­ m - YN .

317


:

3.

§ 13

12.2. (12.4) ­ (12.7) . f H , (12.11)

X


N

= A -1FN N =O N
2, 2 N

N} A -1FN - {ts;; N} A -1F N {

L

(

-

ln N ,

)

(12.13)

12.2.
N} N} A -1FN - {ts;; N} A -1F = A -1 {ts;; N } F - {ts;; N} A -1F = N N { { N} {

=A

-1 N

(

N} {ts;; N } A - A N {ts;; N } A -1 F . { N} {

)

(12.14)

(12.14),
N} A -1FN - {ts;; N } A -1F N { L2


,2N

A

-1 N

N} {ts;; N } A - A N {ts;; N } A -1 F { N} {

L

.
2, 2 N

(12.15)

(12.9), (12.12) (12,15) (12.13). 12.2 .

13. , , , . 13.1. .

A = B - K,

A: H H ,

(13.1)

­ , B ­ , K ­ , H ­ .
318


:

3.

§ 13

Ax = f , (13.2) f H

An xn = f n ,

n .

(13.3)

H n , n ­ , { n }, {

(13.3).
n

}

, n ­

, ,

n : H H n , n : Hn H,

n , n .

(13.4) (13.5)

, (13.4) (13.5) :

n n y = y,
n

y H n , n ,
x H , x H .

(13.6) (13.7) (13.8)

li m n n x - x = 0,
n

li m n x = x ,

, (13.1) , A A [95]:

d i m k e r A = d i m k e r A = k ,
k 0 ­ . :

(13.9)

e1 , . . ., ek ­ k er A ; g1 , . . ., g k k e r A ;

An = n A n ,
319

Bn = n B n .

(13.10)


:

3.

§ 13

, n
d i m k e r An = d i m k e r An = k .

(13.11)

, (a A AN , a E + A AN ) , A

A N , §§ 1, 9, 12 (13.9) (13.11) . :

Gx = Gn y = Gn y =
e

m =1

(

k

em , x ) g m , ,y g

(13.12)

m =1


k

k

( (

e

n, m

) )

n, m

,

(13.13)

e

m =1

n, m

,y g
g

n, m

,

(13.14)

n, m

k e r An , = n em ,

n, m

k e r An ,

(13.15) (13.16)

e

n, m

g

n, m

= n gm ,

n ,


m = 1, . . ., k .
n

{Cn }

C n : H n H

-

(), n0 ­ M , n n0
-1 n +

C

( C n )

C

-1 n

M,

(

C

+ n

M

)

n n0 .
,g

. An , Bn , G , Gn e
320
n, m n, m

, -


:

3.

§ 13

13.1, (13.10), (13.12), (13.13) (13.16). 13.1.

{ Bn }
n, k

-

ke r Bn = {0} .

e

n,1

, .. . , e

n, k

ke r An , g

n,1

, . .. , g

ker An , n n

0

(13.17)

,
n

lim e

n, m

-e

n, m H

= 0, l i m g
n

n

n, m

-g

n, m H

= 0, m = 1, . . . , k , (13.18)
n


n

lim

n

( A + G ) - 1 f - ( An + G

n

)

-1

n f

= 0,
H
n

(13.19)

f H ;
n + l i m n A + f - An n f H

= 0,
n

(13.20)

f R ( A ) . 13.1 . 13.4. An , Gn , f n

f An , Gn , n , An - An n , Gn - Gn n , f n - n f
H

n 0 , n 0 ,
,
n

(13.21) (13.22) (13.23)


n

n

f

nH

n 0 ,

n 0, n 0, n 0 ­ .
321


:

3.

§ 13

13.2. An , G n : H n H f n H
n

n



(13.21) - (13.23)



(

An + G

n

)

-1

= mn m < ,

(13.24) (13.25)

mn n +


(

n

)
f

= qn q < 1 .

(

An + G

n

)

-1

f n - An + G

(

n

)

-1

n


H
n

m n + q 1- q

(

n

)

f

nH

(13.26)
n

n n0 . 13.2 . 13.5. 13.2. . , 13.1 13.2. 13.1. A + G , G (13.12), . 13.1. , A + G A + G . A + G . x0 H

( A + G)

-1



A x0 + G x 0 = 0 .
, x0 = 0 . ,

(13.27)

Gx0 =

m =1

(

k

e m , x 0 ) g m k e r A ,
322

A x0 R ( A ) k e r A , (13.28)


:

3.

§ 13

(13.27)

A x0 = 0,

G x0 = 0 .

(13.29)
k

(13.29) 1 , . . ., ,

-

x0 = 1e1 + . . . + k ek .

(13.30)

(13.29) (13.28)

1 = ( e1 , x0 ) = 0, ... , k = ( ek , x0 ) = 0 .
(13.30) (13.31) x0 = 0 . k e r A + G

(13.31)

(



)

= {0} . . -

[95]. .

G u =

m =1

(

k

g m , u ) em .


(13.32)

, , , A u 0 + G u 0 = 0 ,

u0 = 0 .
13.1 . (13.2)

Ax + Gx = f .


(13.33)

13.2. f R ( A ) . (13.2) (13.33):

A+ f = ( A + G

) -1 f

.

(13.34)

323


:

3.

§ 13

. x0 ­ (13.2), x0 k e r A , em , x x
0

(

0

)

= 0, m = 1, . . ., k , , Gx0 = 0 .

(13.33),

(13.34) x0 ­ (13.33):

A x0 + G x0 = f .


(13.35)

G x0 k e r A A x0 R ( A ) k e r A , (13.35)

A x0 = f ,

G x0 = 0 .

(13.36)

(13.36) , x0 (13.2), , ­ . 13.2 . An (13.7)
n

li m n A x - An n x

H

= 0,
n

x H .

(13.37)

, (13.6) (13.7), x H

n A x - An n x

H

n

= n A x - n n x

H

0, n .

13.3. . [30]: ­ . C

{ n}

: C n : H n H n ,

x n ker C n

324


:

3.

§ 13

Cn x
:

nH

0
n



n n.

(13.38)

x

nH

0
n



(13.39)

13.3.

{ Bn }

-

ke r Bn = {0} .


(13.40)

{

An + G

n

}

­ ,

{ An }

­ .

13.3. e

{ n}

k e r A ­ -

, (13.8) ­ ,
n

li m ne

mH

=e
n

mH

=1



m = 1, . . ., k ,

(13.41)


n li m ( n em , n el ) = li m n n em - em , el = 0 1 m l k . (13.42) n

(

)

(13.41) (13.42) (13.17), (13.18). .

{

An + G

n

}

. -

{

An + G

n

}

. -

xn ,

(

An + Gn ) x

nH

0,
n

n.

(13.43)

(13.18)
n

l i m ( Gn - G

n

)

y

H

= 0,
n

y H n ,

(13.44)

325


:

3.

§ 13

Gn (13.14), (13.15). (13.43) (13.44) ,

(

An + Gn ) x

nH

n

( An + Gn ) x

nH

n

+ ( Gn - Gn ) x

nH

, n.
n

, ,

An x


nH

0,
n

(

e

n, m

,x

n

)

0, n , m = 1, . . ., k .

(13.45)

xn = y n + z n ;

y n k er An , z n k e r An ,

(13.46)

(13.45) (13.46)

An z

nH

0,
n

(

e

n, m

,y

n

)

0, n , m = 1, . . ., k .
n ,1

(13.47) ,

y n k e r An ,

, . . .,

n, k

y n = n,1e

n ,1

+ .. . +

n, k n , k

e

. (13.47)



n, m

=e

(

n, m

,y

n

)

0,

n .

(13.48)

, y n (13.48) y
nH

0, n , ,
n

z

nH

1,
n

n.

(13.49)

n z

{

n

}

. -

K , , z 0 H ,

K n z n 0,

n.

(13.50)

326


:

3.

§ 13

(13.47), (13.50) (13.7) :

Bn z n - n B - 1 z

(

0

)

H

0,
n

n.

(13.51)

(13.51) B

{ n}

k e r Bn = {0}
z n - n B -1 z

0H

0,
n

n.

(13.52)

n A B -1 z

0H

n

An z

nH

n

+ An n B -1 z 0 - z
0H

(

n

)

H

+
n

+ n A B -1 z 0 - n n B -1 z

0,

n.

(13.53)

(13.8) (13.53), ,

A B - 1z 0 = 0 , B -1z 0 k e r A . B -1 z 0 =
m =1



k

mem .

(13.54)

n (13.54),

n B -1 z 0 =

m =1



k

me

n, m

,

n .

(13.55)

(13.18), (13.52) (13.55) ,

zn -

m =1



k

me

n, m H
n

0,
k

n.

(13.56)

,

m =1



me

n, m

k e r An

z n k e r An , (13.56) z
nH

0,
n

n.

(13.57)

327


:

3.

§ 13

(13.57) (13.49).

{

An + G

n

}

.

An . 13.3 . . (13.40) 13.3 . 13.3, (13.40).

{}

{ Bn }



{ An }

,

An + G

{

n

}

.

. H = l 2

B = d i a g B1 , B 2 , . . . , B m , . . . , K = d i a g K 1 , K 2 , . . . , K m , . . . ,


(13.58) (13.59)

0 1 Bm = , 1 0 1 0 - m Km = 0 -1 m ,

m = 1, 2, . . . ,

(13.60)

m = 1, 2, . . . .

(13.61)



H

n



( 2n + 1)

-

2 n +1

: Hn =

2 n +1

, n . -

n n .

x = (1 , 2 , . . . , n , . . . y = (1 , 2 , . . . ,
2 n +1

)

T

l2 , , n .

(13.62) (13.63)

)

T



2 n +1

328


:

3.

§ 13

n n (13.62) (13.63) :

n x = ( 1 , 2 , . . . , n y = (1 , 2 , . . . ,
2 n +1

2 n +1

)

T

Hn , n .

(13.64) (13.65)

, 0 , 0 ,...

)

T

H,

, n n (13.64) (13.65) n : H H n , n : H n H (13.6) ­ (13.8). An B (13.61) :
n

(13.58) ­

1 An = d i a g A1 , A 2 , . . . , A n , , A m = Bm - K m , n +1

(13.66) (13.67)

Bn = d i a g B1 , B 2 , . . . , B n , 0 .
(13.60), (13.66) (13.67) ,
+ Bn = Bn ,
+ + - An = d i ag A1 , A - 1 , . . . , A n 1 , n + 1 , 2

(13.68) (13.69)

+ A1 =

1 4 1 4

1 4 1 4

,

(13.70)

A

-1 m

m m2 -2 m -1 m2 - = m2 m -2 2 m -1 m-

1 , 1

m = 2, . . . n .

(13.71)

, (13.68) B

+ n

= 1 n ,

{ Bn }

, (13.69) ­ (13.71)

329


:
+ An = n + 1 n ,

3.

§ 13

{ An }

-

. 13.3.
13.4. n

{

n

}

-

.

{ An }

B

{ n}
{

.

13.4. = E , E : H H ­ .
, n n

}

( ), x H ,

(

nn

x, x ) ( x, x ) , x, x ) ( x, x ) ,

n.

(13.72)

, (13.8), x H

(

nn

n .

(13.73)

(13.72), (13.73)

( x, x ) = ( x, x )



= .

(13.74)

(13.73), (13.74) [6, . 121] = E . B .

{ n}

. -

{ Bn }

. . -

x

{ n}

,

xn k er B

n



Bn x

nH

0, n .
n

330


:

3.

§ 13

K , , ,

K n xn x0 H ,
(13.75)

n.
n .

(13.75)



n

(

A n x n + x

0

)

H

0,
n

(13.76)

n

{

n

}

-

, x H

(

x, A n xn + x

0

)=(

n x,
0

n

(

A n xn + x

0

))

+
(13.77)

+ x - n n x, A n x n + x

(

)

0,

n.

, (13.77)

{

A n x n + x

0

}

n . -

x0 A . A . . , x0 A .

n x0 - n A n n A + n n x
(13.76) (13.78) ,

0H

0,
n

n.

(13.78)

An xn + n A + x

(

0

)

H

0,
n

n.

(13.79)



{ An }

, ­ -

(13.38), (13.39), (13.79),

xn + n A + x0 + y

nH

0,
n

n,

(13.80)

331


:

3.

§ 13



y n k e r An , x n + n A + x0 + y n k er An .
, y

(13.81) (13.80) -

{ n}

{ xn } { n }
.

.
n, k n, k

y n = n,1e


n ,1

+

n , 2 n, 2

e

+ .. . +

e

(13.82)



n, m

c o n s t < ,

m = 1, . . ., k , n .

(13.83)

(13.83), , ,




n, m



m



n , m = 1, . . ., k .

(13.84)

y 0 = 1e1 + 2e2 + . . . + k ek .
(13.81), (13.82), (13.84) (13.85)

(13.85)

n y0 - y
xn +
Bn x

nH

0,
n

n.

(13.86)

(13.80) (13.86)
n

(

A + x0 + y

0

)

H

0,
n

n.

(13.87)

nH

0 n (13.8)
n

B A + x0 + y

(

0

)

= 0 . ,
n

A + x0 + y 0 = 0 . n
(13.87) , x xn . 13.4 .
332
nH
n

(

A + x0 + y

0

)

= 0 .

0 n . -


:

3.

§ 13

13.4. 13.1.

en,1 , e

n, 2

, . .. , e

n, k

k er An ,

g n ,1 , g

n, 2

, . .. , g

n, k

k e r An

13.3. (13.19).

n x f - ( An + G
( An + G
x f = An + G

n

)

-1

n f


H
n

n

)(
-1

An + Gn ) n x f -

n

(

An + Gn ) x

fH

,
n

(

n

)

-1

f , (13.37), 0,
n



(


Gn n - nG ) x

fH

n.

(13.88)

(

G n n - n G ) x f =

m =1



k

(

n n em - em , x

f

)
{

g

n, m

.

(13.89)

(13.8) n

n

}

-

. (13.89) (13.88). (13.19) . (13.20) (13.19) 13.2. 13.1 . 13.5. 13.2. -

An + G n

(

An + G

n

)

-1 H
n



m . 1- q

(13.90)

:

Dn = An + Gn ,
333

Dn = An + Gn .

(13.91)


:

3.

§ 13



- Dn = E n - ( Dn - Dn ) Dn 1 Dn ,

(13.92)

E n ­ H n . (13.91) (13.24), (13.25)

(

- Dn - Dn ) Dn 1 Dn - Dn D

-1 n

( n +

n

)

mn q < 1 ,

(13.93)

mn =

(

An + G

n

)

-1

. (13.93)

E n - ( Dn - Dn ) D

-1 n

[42]
- E n - ( Dn - Dn ) Dn 1 -1



1 . 1- q

(13.94)

(13.92) (13.94) Dn
(13.90).
- - Dn 1 f n - Dn 1 f

n

D

-1 n

fn - f

n

+D

-1 n

-D

-1 n

f

n

.

(13.95)

D

-1 n

- - Dn 1 .

D

-1 n

-D

-1 n

=D

-1 n

(

- Dn - Dn ) Dn 1 ,

(13.90)

D

-1 n

-D

-1 n



mq . 1- q

(13.95) (13.26). 13.2 .

334


:

4.

§1

4
§ 1. :

dx = Ax + f ( t ) , dt
-, C N = C

0 t 2 ,
N

( 1 .1 ) ­ -

A ­ N , f ( t ) C
N

[

0, 2

]

­ 2 -

-. 2 - (1.1) . , A , k

d e t ( A - i k E ) 0,

k .

(1.2)

A , . (1.1)

B ­ N , ­ ­ . 2 - (1.1). (1.1)

dx = ( A + B ) x + f (t ) , dt

0 t 2 ,

(1.3)

x (t

)

x ( t ) = e pt C + e
C ­ .

pt


0

t

e

- ps

f ( s ) ds ,

335


:

4.

§1

(1.1) - f ( t ) C N . [28]. (1.1) C
N

- f ( t ) C

N



A . (1.1)

x0 ( t ) = e

tA

(

E -e

2 A -1

) t



2

e

( 2 - s ) A

f ( s ) ds + e
0

t

- sA

f ( s ) ds .

(1.4)

, (1.1) f ( t ) C

N

2 - x ( t ) . A . (1.1)

x ( t ) = e t AC + e

tA


0

t

e

- sA

f ( s ) ds ,

(1.5)

C ­ . 2 -

(

E -e

2 A

)

x (0) = e

2 A

2


0

e

- sA

f ( s ) ds .

(1.6)

x ( 0 ) , (1.6), x ( t ) C
N

(1.1).

, A , 1 - e 2 E - e 2 A . ( = i k , k e
2 A

)

A , 1

. (1.6),
336


:

4.

§1

-

x ( 0 ) , , . 2 (1.1). A . . A , 1 e 2 A . (1.6) 2 - -

f ( t ) . 2 - -

(1.1). . , (1.2). , (1.3) A . (1.3) B . , B = 0 , (1.1) f ( t ) C N . 2.1 A . A . (1.5) (1.3):

x ( t ) = e

t ( A+ B

)

C +e

t ( A+ B

)


0

t

e

- s ( A+ B

)

f ( s ) ds .

( 1 .7 )



(

E -e

2 ( A+ B

)

)

x ( 0) = e

2 ( A+ B

)

2


0

e

- s ( A + B

)

f ( s ) ds .

(1.8)

, B 0 > 0 , : 0 < < 0

det E - e

(

2 ( A+ B

)

)

0.

(1.9)

337


:

4.

§1

(1.9)

d e t ( A + B - i k E ) 0,

k .

d e t ( A - i k E ) 0 k ,

: 0 < < 0
A + B - i kE = ( A - ikE ) E + ( A - i kE
0 > 0

(

) -1 B

)

.

0 ( A - i k E ) B < 1,


-1

k .

(1.10)

:0< <

0

(1.11)

(2.7). , x ( 0 )

A + B - i kE k

(1.8) . , , 2 - (1.3).

x - x0 M ,
, .

(1.12)

x0 x (1.4) (1.7), M > 0 ­ -

, , A , (1.1) (1.3)

> 0 . 2 -

, , (1.4) (1.7). (1.7) , (1.10) (1.11).
0

338


:

4.

§1

(1.12), x0 ( t

)

x ( t ) -

x0 ( t ) = ( a1) ( a 2


) [ ( a3 ) + ( a 4 ) ]

,

x ( t ) = ( b1) ( b 2

) [ ( b3 ) + ( b 4 ) ]

,

( a1) = e tA , ( b1) = e
( a2) = ( E - e
( a 3) = e
0 2 2 A - 1

t ( A + B

)

,
) -1

)

,

( b2 ) = ( E - e
( b3 ) = e
0 2

2 ( A+ B

)

,

( 2 - s ) A

f ( s ) d s, f ( s ) d s,

( 2 - s )( A+ B )

f ( s ) ds ,

( a4) = e
0

t

- sA

( b4 ) = e
0

t

- s ( A+ B

)

f ( s ) ds ,

, x ( t ) - x0 ( t :

)



x ( t ) - x0 ( t ) = [( b1) - ( a1)] ( a 2
+ ( a1) [( b 2 ) - ( a 2 + ( a1) ( a 2

) [ ( a 3) + ( a 4 ) ] +

) ][ ( a3 ) + ( a 4 ) ] +

){[( b3) - ( a3)] + [( b4 ) - ( a 4 )]} +

+ [( b1) - ( a1)

][( b2 ) - ( a 2 )][( a3) + ( a 4 )]

+

+ [( b1) - ( a1)] ( a 2 + ( a1) [( b 2 ) - ( a 2 + [( b1) - ( a1)

){[( b3) - ( a3)] + [( b4 ) - ( a 4 )]} +

)]{[( b3) - ( a3)] + [( b4 ) - ( a 4 )]} +
.

][( b2 ) - ( a 2 )]{[( b3) - ( a3)] + [( b4 ) - ( a 4 )]}

(1.10), (1.11)
339


:

4.

§1

( b1) - ( a1) M 1 , ( b3) - ( a3 ) M 3 ,


( b2 ) - ( a 2 ) M 2 , ( b4 ) - ( a 4 ) M 3 ,
M 3 = 2 M

M 1 = 4 B e

2 A

,

M 2 = m1m2 M 1 ,
,

1

f

C

,
N

m1 = E - e
f
C

(

2 A - 1

)

m2 = s u p
kC

0< <

(
0

E -e

2 ( A+ B

) -1

)

,

= max f
N

1 k N

,

f

kC

= max f k ( t ) .
0 t 2

, : 1.1. (1.3) B , f ( t ) C
N

-

, (1.11), C N , A . 1.2. A . 2 - 0 2 ,

x - x0 ( 1 + 1


)(

2 + 2

)(

3 + 3 ) - 1 2 3 ,

= max ( a1) , = M , 1 1 1 0 t 2 2 = 0max ( a 2 ) , 2 = M 2 , t 2 3 = 41 f C , 3 = 2M 3 . N

340


:

4.

§2

2. . , . B , . : 1) , ; 2) , ; 3) , , . , A . 1). 2) 3) § 3 § 4, . .

dx = px + f (t ) , dt

0 t 2 ,

( 2 .1 )

p = c o n s t , f ( t ) ­ 2 -

f ( 2 ) = f ( 0 ) .
(2.1) 2 - (2.2):

( 2 .2 )

2 - x ( t

)

x ( 2 ) = x ( 0 ) .

( 2 .3 )

(2.1)

x ( t ) = C e pt + e
C ­ .

pt


0

t

e

- ps

f ( s ) ds ,

( 2 .4 )

341


:

4.

§2

(2.4) 2 - , , C

(1

-e

2 p

)

C =e

2 p

2


0

e

- ps

f ( s ) ds .

(2.5)

, x ( t ) 2 - , (2.4) t = 0 t = 2 , (2.3), (2.5). , (2.5). (2.4) ,

x ( 0 ) - x ( 2 ) = 1 - e

(

2 p

)

C -e

2 p

2


0

e

- ps

f ( s ) ds = 0 .

(2.3). , x ( t ) ­ 2 - . 1 - e C :
2 p

0 , (2.5) C= e 2 p 1 - e 2
2 p


0

e

- ps

f ( s ) ds = 0

2 - (2.1)

x0 ( t ) =

e p ( t + 2 1 - e 2

) 2
p


0

e

- ps

f ( s ) ds + e

pt


0

t

e

- ps

f ( s ) ds .

(2.6)

1 - e 2 p = 0 , (2.5)
2


0

e

- ps

f ( s ) ds = 0 .

(2.7)

(2.7) (2.1) 2 - (2.4), C ­ .
342


:

4.

§2

, 1 - e

2 p

0
(2.8)

p ik ,
1 - e p = i k :
2 p

k ,

= 0 k , -

p = ik ,

k .

(2.9)

, 2 - (2.1) , (2.8), , (2.9). (2.1)

dx = ( p + ) x + f (t ) , dt

(2.10)

0 ­ , . , p ( p ) > 0 , ,

: 0 < < ( p) ,


(2.11)

p + ik ,

k .

(2.12)

(2.12) (2.8)

2 - (2.1).
(2.11). (2.10) 2 -

x ( t ) =

e ( p + )(t 1 - e 2 (
( p + )t

+ 2 p +

) 2 )


0

e

-( p + ) s

f ( s ) ds +

+e


0

t

e

-( p + ) s

f ( s ) ds .

(2.13)

343


:

4.

§2

x ( t ) (2.8) (2.9).

0

(2.8). (2.13) = 0 , ,

x - x0 0,
x0 ( t

0,

)

(2.6)

2 - (2.1).
(2.9) k . 2 - , (2.1), (2.7).

x

00

(t ) =

e pt 2

2


0

se

- ps

f ( s ) ds + e

pt


0

t

e

- ps

f ( s ) ds ,

(2.14)



x - x

00

0,

0.
2 p

(2.15)

, (2.9) e

= 1 . 0 -1

x ( t ) = e

( p + )t

2 1 2 e - 1 2
( p + )t

2


0

se

- ps

e

- s

- s

f ( s ) ds +

+e


0

t

e

-( p + ) s

f ( s ) ds .

(2.16)

(2.16) (2.14) (2.15). , x (2.1).
00

(t )

2 - -

344


:

4.

§2

, (2.7),

x

00

( 2 ) =

e 2 2 =

p 2


0

se

- ps

f ( s ) ds + e

2 p

2


0

e

- ps

f ( s ) ds =

1 2

2


0

se

- ps

f ( s ) ds = x

00

(0)

.



2 -

x

00

(t )

. , x

00

(t )
2

(2.1):
- ps

x 0 ( t ) = p 0

e pt 2


0

se

f ( s ) ds + pe

pt


0

t

e

- ps

f ( s ) ds +

+ e pt e

- pt

f ( t ) = p x 0 ( t ) + f ( t ) . 0

, ( ):

x ( t ) , li m x ( t ) = 0 0 x00 ( t ) ,

p i k , k , p = i k , k .

x ( t ) 0 . :

x - x
(2.8);

0 C[ 0, 2

]

m0 ,

(2.17)

x - x

0 0 C [ 0 , 2

]

m0 0 ,

(2.18)

(2.9).

m0 = ( m1m2 + 1) ( m3m4 + 1)
345

2

g (s)
0

ds ,


:
2

4.

§2

m

00

= ( 8 + 2 ,

) g ( s ) ds
0

,

m1 = e m3 = s u p

2 p

m2 = 1 - e
) -1

(

2 p - 1

)

,
( t - s )( p + )

0< <

(1
0

-e

2 ( p +

)

, m4 = s u p e
0< < 0 0 s , t < 2

.

, : C [ 0, 2 2.1. ) (2.1)

]

f ( t ) C [ 0, 2 ] , -

(2.8). ) (2.9). (2.1) C [ 0, 2

]

f ( t ) C [ 0, 2

]

-

(2.9). 2.2. (2.11). : ) f ( t ) C [ 0, 2

]

(2.10)

2 - (2.13), (2.17), (2.8); ) f ( t ) C [ 0, 2 ] , (2.7), (2.10) 2 - (2.13), (2.18), (2.9).

346


:

4.

§3

3.

dx = Ax + f ( t ) , dt


0 t 2 ,

( 3 .1 )

f ( t ) = ( f1 ( t ) , ... , f

m

(t )) C
T T

m

­ 2 - -,

x ( t ) = ( x1 ( t ) , ... , x

m

(t )) C

m

­ 2 - -,

0 p A= 0 p

, p = con st; A : C C m
m

m

(3.2)

­ , C

­ m - -

-. m = 1 (3.1) (2.1), . m > 1 . (3.1) (3.2) m (2.1):

0 p A= 0 p

, p = con st; A : C C m

m

(3.3)

, (2.8) (3.3) C = C [ 0, 2 k = 1, . . ., m .

]

347


:

4.

§3

(2.8). (3.3)





2 -

x

k,0

(t ) =

e p( t + 2 1 - e 2

) 2
p


0

e

- ps

f

k

( s ) ds + e

pt


0

t

e

- ps

f

k

( s ) ds

.

(3.4)

(2.8), (2.9) k . (3.3) k
2


0

e

- ps

f

k

( s ) ds = 0

.

(3.5)

, (3.3) 2 - . (3.1).

dx = ( A + E ) x + f (t ), dt

0 t 2 .

(3.6)

(3.6) m (2.10):

dx

k

dt

= ( p + ) x k + f k ( t ) , 0 t 2 , k = 1, . . ., m .

( p ) > 0 , : 0 < < ( p ) (3.6) f ( t ) C :
m

2 - -

x ( t ) = x1,


(



( t ) , ... , x
f
k

m ,

(t )) C
T

m

,
t
-( p + ) s

(3.7)

x

k ,

(t ) =

e

( p + )( t + 2 ) 2
2 ( p +

1- e

)


0

e

-( p + ) s

( s ) ds + e

( p + )t


0

e

f

k

( s ) ds

.

348


:

4.

§3

, 2 - - C
m

,

x

C

= max x
m

1 k m

kC

x

kC

= ma x x k ( t ) .
0 t 2



x0 ( t ) = x1, 0 ( t ) , ... , x
x1,
0

(

m, 0

(t )

)

T

,

x

00

( t ) = ( x1, 00 ( t ) , ... , x

m, 00

(t )

)

T

,

( t ) , ... , x
2

m, 0

(t )
k

(3.4),

x

k , 00

(t ) =

e pt 2


0

se

- ps

f

( s ) ds + e

pt


0

t

e

- ps

f

k

( s ) ds, k = 1,. .., m

.

2.2, , x ( t ) 0 :

x - x
(2.8);

0C

m

M0 ,

(3.8)

x - x
(2.9). M m0 m
00 0

00 C

M
m

00

,

(3.9)

M

00



(2.17) (2.18), .

, , 2.1 2.2: 3.1. ) (3.1) C
m

- f ( t ) C

m



(2.8).

349


:

4.

§4

) (2.9). (3.1)

C

m

-

f ( t ) C m (3.5).
3.2. (2.11). : ) - f ( t ) C
m

(3.6)

2 - (3.7), (3.8), (2.8); ) - f ( t ) C m , (3.5), (3.6) 2 - (3.9), (3.9), (2.9).

4. : 1) , ; , ( ); 2) , m 2 m -; , m ( ); 3) , m 2 ; m 1. § 2 § 3 . 1) 2). 3).
350


:

4.

§4



dx = Ax + f ( t ) , dt


0 t 2 ,

( 4 .1 )

0 p 1 , A= 1 p 0

p = con st .

(4.2)

(4.2) [25]. (4.1) - f ( t ) C
m

C

m

(2.8). :

x0 ( t ) = x1, 0 ( t ) , ... , x


(

m, 0

(t )
pt

)

T

,

x

k,0

(t ) =

e p( t + 2 1 - e 2

) 2
p


0

e

- ps

g

k

( s ) ds + e

2


0

e

- ps

g

k

( s ) ds

.

g

k

(t )

-

f k (t ), g k (t ) = f k (t ) + x

k = 1,
k -1, 0

( t ) , k = 2,. . ., m.

.

(2.8), (2.9). (4.1) - f ( t ) C
2 m

(4.3)


0

e

- ps

f1 ( s ) d s = 0 .

(4.3) (4.1) C m .

x

00

( t ) = ( x1, 00 ( t ) , ... , x
351

m, 00

(t )

)

T

,


:

4.

§4



x

k , 00

(

- t) = -

l1 = 0

I ( l1 ) + J ( l2 ),
l2 =0 k -1 k

k

k -1

k = 1, . . ., m - 1, k = m,



l1 =1

I ( l1 ) +
2

l2 =0

J ( l2 ),
)

I ( l1 ) = J (l

1 e 2 l1 ! 0

ik (t - s

( + t - s ) l f
1

k +1-l1

( s ) ds
.

,

2

)

1 = e l2 ! 0

t

ik ( t - s

)

(t - s )l f
2

k +1-l

2

( s ) ds

(4.1)

dx = ( A + B ) x + f (t ) , dt

0 t 2 ,

(4.4)

B ­ m .

f ( t ) = ( f1 ( t ) , ... , f

m

(t )) C
T

m

.

(4.4) 2 - . . m

1 1 -1 -1 w1 1 w0 = m - m +1 m -1 - m +1 m -1 w0 w1 . (4.5)
1 F= m 1 w0
m w0 -1

... ... ...
-1

1

wm

- m +1

-1

w

m -1 m -1

,

(4.5)

, = 1

1 w1 w1m -1
352

... 1 ... wm -1 , m -1 ... wm -1


:

4.

§4

m . (4.5) 0 ,





-1

1 w0 ... 1 1 w1 ... = m 1 wm -1 ...
1 D = 0

m -1 m

m

m w0 -1 -1 m -1 w1 , -1 m -1 wm -1

,
-1

0 , - m +1

(4.6)



= D F .
F
-1

(4.7)

= F D 0 , (4.7)





-1

= F D-1 .
: 0 < < 0 ,

B0 , (4.4) f ( t ) C (4.4) 0 . B0 , , . , , , 1
m

0 ­ ,

0 ... 0 1 0 ... 0 0 , B0 = 0 ... 0 0
.
353

(4.8)


:

4.

§4



0 ... 0 1 0 A0 = 1 0
(4.2) p = 0 .

0 0 0

,

(4.9)

. 4.1. 0 ­ =
1/ m

.

D-1 ( A0 + B0 ) D = P ,
0 ... 0 1 0 P= 1 0
1 0 0





­ m . 4.1 . 4.2. 0 ­ =
1/ m

. (4.10)





-1

(

A0 + B0 ) = d i a g ( w0 , w1 , . . . , wm

-1

)

,

A0 B0 (4.8) (4.9). ,

F P F = d i a g ( w0 , w1 , . . . , w
4.1 (4.10).

m -1

)

354


:

4.

§4

A + B0 , A (4.2). 4.3. 0 ­ =
1/ m

.





-1

(

A + B0 ) = d i a g ( p + w0 , . . . , p + wm

-1

)

,

(4.11)

A B0 (4.2) (4.8). ,

A + B0 = ( A0 + B0 ) + p E







-1

( pE ) = pE

,

(4.12)

(4.10) (4.11). (4.4)
- d 1x

dt


=



-1

(

- - A + B0 ) 1 x + 1 f ( t ) .

(4.13)

- y = 1 x, - g (t ) = 1 f (t ) ,

y = ( y1 , . . . , y

m

)

T

,

(4.14)

g ( t ) = ( g1 ( t ) , . . . , g

m

(t )

)

T

(4.15)

(4.12), (4.13) :

dy

k

dt

= ( p + wk

-1

)

y k + g k ( t ) , k = 1, . . ., m .

(4.16)

p ( p ) > 0 ,

0< < ( p


)



0<

m

< ( p

)

(4.17)

355


:

4.

§4

p + wk
C

-1

0,

k = 1, . . ., m .

2.1. ) (4.16)
m

- g ( t ) C m :
2

yk (t ) =



0

k ( t , s ) g k ( s ) d s + k (t , s ) k ( s ) ds ,
0

t

(4.18)

k (t, s ) =

e

(

p + wk

-1

)

( 2 + t - s )
-1

1- e

2 ( p + wk

)

,

k (t, s ) = e(

p + wk

-1

)

(t -s )

,

k = 1, . . ., m .

(4.14),

x ( t ) = y ( t ) = ( x1 ( t ) , . . . , x

m

(t )

)

T

.

, (4.15) (4.5),

xn ( t ) = g k (t ) =

1 m


j =1

m



- n +1

w

n -1 j -1

y j (t ),

n = 1, . . ., m ,

(4.19)

1 m


l =1

m



l +1

l wk+1 f l ( t ) , -1

k = 1, . . ., m .

(4.20)

(4.18) (4.19) , (4.20), 2 - (4.4):

x

n,

(t ) =

1 m
m

2 m 0

j =1 l =1

m



-( n -l - 2 ) / m

w

n -l - 2 j -1

j ( t , s ) f l ( s ) ds +

+

1 m

2 m 0

j =1 l =1



-( n -l - 2 ) / m

w

n -l - 2 j -1

j ( t , s ) f l ( s ) d s, n = 1, . . ., m . (4.21)

, (2.17) (2.18),

356


:

4.

§4

x - x
(2.8);

0C

m

M0 ,

(4.22)

x - x

00 C

M
m

00

,

(4.23)

(2.9) (4.3). M 0 M
00

­ , .

. 4.1. ) (4.1) C
m

- f ( t ) C

m



(2.8). ) (2.9). (4.1)

C

m

-

f ( t ) C m (4.3).
4.2. (4.17). : ) - f ( t ) C
m

(4.4)

2 - (4.21), (4.22), (2.8); ) - f ( t ) C m , (4.3), (4.4) 2 - (4.21), (4.23), (2.9).

357


: .


() . , , . . - . . , . . , . . , . , , , , , , , , , , . , . , , . - (. . : [. . . . . . 1982. . 20. ­ . 117 ­ 178].

358



1. , . . / . . . ­ .: , 1979. - 512 . 2. , . . / . . // . ­ 1979. ­ . 22, 2. - . 71 ­ 74. 3. , . . / . . , . . // . ­ .: . 1986. 73. - 21 . 4. , . . / . . , . . // . . ­ 1986. - 8. - . 3 ­ 9. 5. , . . / . . // . . ­ 1988. - 3. - . 3 ­ 8. 6. , . . / . . , . . . ­ .: , 1966. - 544 . 7. , . . / . . // . - 1967. ­ . 7, 3. - . 672 ­ 676. 8. , . . / . . . ­ .: , 1968. - 90 . 9. , . . / . . // . 1968. ­ . 8, 2. - . 426 ­ 428. 10. , . . / . . // . ­ .: , 1969. - . 12. - . 56 ­ 79.. 11. , . . / . . , . . . ­ .: , 1989. - 199 . 12. , . . / . . . ­ .: , 1983. - 336 . 13. , . . / . . , . . , . . // . - 1983. ­ . 47, . 5. - . 781 ­ 789. 14. , . . / . . , . . . ­ .: . 1985. - 256 . 15. , . . / . . , . . . ­ .: , 1962. ­ 1. - 464 . 16. , . . , / . . // . - 1980. ­ . 20, 1. - . 38 ­ 50. 17. , . . / . . , . . // . 1968. ­ . 6, 4. - . 27 ­ 37.

359


18. , . . / . . , . . // . ­ 1974. ­ .. 215, 5. ­ . 1032 ­ 1034. 19. , . . / . . // . - 1977. ­ . 17, 4. - . 847 ­ 858. 20. , . . / . . // . - 1972. ­ . 12, 1. - . 207 ­ 212. 21. , . . / . . , . . // . ­ .: , 1983. ­ . 1. - . 124 ­ 266. 22. , . . / . . . ­ .: , 1980. - 400 . 23. , . . / . . , . . . ­ .: , 1987. - 320 . 24. , . . . . / . . // . ­ .: , 1977. . 4. - . 21 - 25. 25. , . . / . . . ­ .: , 1966. - 576 . 26. , . . / . . // . - 1972. ­ . 12, 6. - . 1572 ­ 1594. 27. , . . - . . / . . // . . - 1978. - . 15 ­ 31. 28. . . . .: . 1967. ­ 472 . 29. , . . / . . , . . . ­ .: , 1970. - 664 . 30. , . / . // . ­ 1977. ­ . 20, 7. - . 3 ­ 6. 31. , . / . // . ­ 1982. ­ . 25, 10. - . 584 ­ 587. 32. , . . / . . , . . // . ­ 1965. ­ . 6, 3. - . 499 ­ 508. 33. , . / . . ­ .: , 1964. - 430 . 34. , . . . 1 / . ­ .: . 1965. - 616 35. , . . / . . // . . ­ 1958. ­ 3. ­ . 99 ­ 106. 36. , . . / . . // . ­ 1962. ­ .. 145, 2. ­ . 270 ­ 272. 37. , . . / . . // . ­ 1962. ­ .. 142, 5. ­ . 997 ­ 1000. 360


38. , . . / . . // . - 1966. ­ . 6, 6. - . 1089 ­ 1094. 39. , . . / . . , . . , . . . ­ .: , 1978. - 206 . 40. , . . . . / . . // . - 1967. ­ . 22, 2. - . 168 ­ 175. 41. , . . / . . , . . . .: , 1984. - 751 . 42. , . / . . .: , 1972. 740 . 43. , . . / . . , . . . .: , 1968. - 496 . 44. , . . / . . . .: , 1971. - 104 . 45. , . . / . . . .: , 1975. - 432 . 46. , . . / . . // . ­ 1955. ­ .. 102, 2. ­ . 205 ­ 206. 47. , . . / . . // . ­ 1956. ­ . 110, 3. ­ . 389 ­ 390. 48. , . . / . . . : . 1962. - 92 . 49. , . . ­ / . . . : . 1973. - 72 . 50. , . . / . . , .. , . . . : . 1980. - 288 . 51. , . . / . . . : , 1981. - 343 . 52. , . . / . . // . ­ 1980. ­ . 255, 5. ­ . 1046 ­ 1050. 53. , . . / . . , . . // . . . . ­ 1986. ­ . 1313. - . 465 ­ 475. 54. , . . / . . , . . // . ­ .: , 1990. ­ . 7. - . 94 ­ 273. 55. , . . ( , , ). .: «», 1995. ­ 520 . 56. , . . / . . // . ­ 1976. ­ . 19, 19. - . 106 ­ 121. 57. , . . L - / . . // . ­ 1979. ­ 5. - . 921 ­ 935. 361


58. , . . / . . // . - 1965. ­ . 6, 1. - . 170 ­ 175. 59. , . . . . / . . // . - 1973. ­ . 13, 5. - . 1099 ­ 1111. 60. , . . / . . . ­ .: , 1974. - 360 . 61. , . . L - / . . // . ­ 1977. ­ . 233, 2. - . 291 ­ 294. 62. , . . / . . , . . // .­ .: . 1982. - . 11 ­ 18. 63. , . . L - / . . . . // : , , . ­ .: . 1983. - . 20 ­ 29. 64. , . . . ­ .: , 1968. - 512 . 65. , . . / . . // . ­ 1979. ­ . 22, 10. 66. , . . / . . // . ­ 1981. ­ . 24, 4. 67. , . . / . . // . ­ .: , 1982. 68. , . . / . . , . // . ­ 1983. ­ . 26, 4. 69. , . . L - / . . , . . // : , , . ­ .: . 1983. 70. , . . / . . // . ­ : . 1983. 71. , . . / . . // . ­ .: . 1984. 72. , . . / . . // . ­ 1984. ­ . 27, 8. 73. , . . / . . // . ­ : . 1984. 74. , . . / . . , . . // . ­ 1985. ­ .. 286, 3.

362


75. , . . / . . // . ­ 1985. ­ .. 28, 6. ­ . 322 ­ 325. 76. , . . / . . // . ­ .: . 1985. 77. , . . / . . , . . // . ­ .: . 1985. 78. , . . / . . , . . // : , , . ­ .: . 1985. 79. , . . / . . , . . // . - 1986. ­ . 8, 6. 80. , . . / . . , . . // . 1986. ­ . 26, 9. 81. , . . / . . , . . // . ­ 1987. ­ .. 30, 11. 81. , . . / . . , . . // . ­ .: , 1987. 82. , . . / . . // . 1989. 4207 ­ 89. ­ 58 . 83. , . . / . . , . // . ­ : , 1989. 84. , . . / . . , . // . ­ . 1991. 85. , . . / . . // , . ­ . 1993. 86. , . . / . . // . ­ .: . 1995. 87. , . . / . . // . ­ . 1998. 88. , . . / . . // . ­ . 2000. 363


89. , . . / . . // . ­ . 2000. 90. , . . / . . , . . // , , . ­ : . 2005. 91. , . . / . . , . . , . . // , , . ­ : . 2005. 92. , . . / . . , . . // , , . ­ : . 2006. 93. , . . / . . , . . , . . // . - 2007. ­ . 47, 12. 94. , . . / . . , . . , . . // . ­ 2008. ­ .. 419, 4. 95. , . . / . . // . . ­ 1943. ­ .. 7, 3. ­ . 147 ­ 163. 96. , . / . . ­ .: , 1976. - 320 . 97. , . . / . . . .: , 2001. - 384 . 98. , . . / . . , . . // . ­ 1976. ­ .. 228, 5. ­ . 1049 ­ 1052. 99. , . . / . . , . . // . ­ 1976. ­ .. 20, 5. ­ . 747 ­ 752. 100. , . . / . . , . . // . ­ 1980. ­ .. 16, 11. ­ . 2039 ­ 2049. 101. , . / . . . .: , 1981. - 158 . 102. , . . / . . // . ­ 1943. ­ .. 39, 5. ­ . 195 ­ 198. 103. , . / . . // . ­ 1963. ­ .. 151, 3. ­ . 501 ­ 504. 104. , . / . . , . . . .: , 1979. - 288 . 105. , . / .. , . . , . . , . . . .: , 1983. - 200 . 364


106. , . . / . . , . . // . ­ 1976. ­ .. 17, 2. ­ . 402 ­ 413. 107. , . . , / . . . ­ .: , 1989. - 184 . 108. , . . / . . // . - 1965. ­ . 5, 5. - . 907 ­ 911. 109. , . . / . . // . .-. . . - 1980. ­ 3. - . 3 ­ 10. 110. , . . / . . // . - 1981. ­ . 17, 8. - . 13 ­ 17. 111. , . . L - / . . // . - 1983. ­ . 23, 3. - . 536 ­ 544. 112. , . . ­ / . . // . - 1985. ­ . 282, 4. - . 804 ­ 808. 113. , . / . . // . ­ 1980. ­ .. 252, 4. ­ . 810 ­ 813.

365


: .


........................................................................ 3 9

1..............................................................
§1 .......................................
1.1. . 1.2. . 1.3. . 1.4. . 1.5. . 1.6. . 1.7. N - . 1.8. . 1.9. . 1.10. . 1.11. .

§2

.....................
2.1. . 2.2. . 2.3. . 2.4. . 2.5. . 2.6. . 2.7. . 2.8. .

25

§3

......................................
3.1. . 3.2. . 3.3. .

33

§4


4.1. . 4.2. . 4.3. 4.2. 4.4. 4.1.

37

§5 §6 §7

. . .............................. ................................................................... ...........................
7.1. . 7.2. 7.1-7.3 (82).

64 69 73

§8

.........
8.1. . 8.2. . 8.3. . 8.4. 8.1.

76

§9 § 10

......................................................
10.1. (). 10.2. . 10.3. . 10.4. . 10.5. 10.1. 10.6. 10.2-10.4.

90 93

§ 11

............................. 366

126


: . § 12 ......................................
12.1. . 12.2. . 12.3. . 12.4. 12.1 ­ 12.4.

139

2..............................................................
§1 L -...............................................
1.1. (). 1.2. L -. 1.3. . 1.4. L -. 1.5. . 1.6. 1.1 1.2.

162 162

§2

L -............................
2.1. . 2.2. . 2.3. () . 2.4. . 2.5. .

184

§3

L - ................................
3.1. . 3.2. . 3.3. 3.1.

202

§4

. . ...............................................................

218 225 225 227

3..............................................................
§1 §2 . ..................................................................... .......................................
2.1. . 2.2. . 2.3. . .

§3

...............................................
3.1. . 3.2. .

237

§4 §5 §6 §7 §8 §9

...................................... .................. ....... ................................ 7.1 ­ 7.4.......................................... ......
9.1. . 9.2. .

243 250 254 262 268 272

§ 10

........................
10.1. b0 + c0 0 , k 0 . 10.2. b0 + c0 = 0 , k 0 . 10.3. b0 + c0 0 , k = 0 .

286

367


: . § 11 § 12 § 13 ....... ........................ ...
13.1. . 13.2. . 13.3. . 13.4. 13.1. 13.5. 13.2.

305 314 318

4..............................................................
§1 §2 §3 §4 .......................... ......................................................... ............................................................. .........................................................

335 335 341 347 350 358 359

......................................................... ..........................................................

368