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ON THE ERROR ESTIMATE OF APPROXIMATION OF FUNCTIONS OF BOUNDED VARIATION BY SZASZ-MIRAKYAN OPERATORS Singh S. N. SKM University, Dumka at Jamtara College, Jamtara, Jharkhand-815351 (India). The Szasz-Mirakyan operators play an important role in the theory of approximation. They have been studied intensively in connection with different branches of analysis. The Szasz-Mirakyan operator is defined as Sn(f, x) = f(k/n) pk(nx), where k=0 pk(nx) = e-nx(nx)k/k!, n N, xR0. The Szasz-Mirakyan operators Sn are defined in terms of a sample of given function f on the points k/n, for k N0, n N. Many research papers [ 3, 4, 5 ] appear with certain modifications in this operator Sn(f, x). Grof [1] proved that if f be continuous on [0, ) and f(x) = O(ex), for some > 0, as x then for all A > 0 and x [0, A] Sn(f, x) - f(x) = O ( 2A(f, n-1/2)), where A(f, ) = sup {| f(x+t) - f(x)| : | t| }. This result was further improved by Hermann. He proved that the above result holds if f(t)=O(tt), > 0. Cheng [1] estimated the rate of convergence of Sn(f, x). He proved that if f be continuous function of bounded variation on every finite subinterval of [0, ) and f(t) = O(tt) for some > 0 as t , then if x (0, ) is irrational, then for n sufficiently large, n x+x/k |Sn(f, x) - (1/2)[f(x+0)+f(x-0)]| ((3+x)/nx) V (gx) + O(x-1/2/n1/2) |f(x+) ­ f(x-)| x-x/k k=1 + O(1) (4x)4x (nx)-1/2(e/4)nx, where Va(g) is the total variation of g on [a, b], and gx(t) = f(t)-f(x+0), x < t < ; = 0 if t = x; = f(t)-f(x-0) if 0 t < x. We shall also consider the continuous functions of bounded variation defined on [0, ) and find the error estimate of approximation by SzaszMirakyan operators maintaining its original form with a different approach, also a better estimate of approximation has been obtained in this paper. References: 1. Grof, J., A Szasz Otto-fele operator approximacics tulajdonsagairol Mat. III, Oszt. Kozl. 20(1971), 35-44. [Hungarian]. 2. Cheng, F., On the rate of convergence of the Szasz-Mirakyan operator for functions of bounded variation, J. Approximation Theory 40 (1984), 226-241. 3. Lehnhoff, H. G., On a modified Szasz-Mirakyan operator, J. Approximation Theory, 42(1984), 278-282. 4. Herzog, M., Approximation theorems for modified Szasz-Mirakyan operators in polynomial weight spaces, Matematiche (Catania), 54 (1999), no. 1 (2000), 77-90. 5. Walezak, Z., On the rate of convergence for some linear operators, Hiroshima Math J. 35(2005),115-124.
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