Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dfc/2014/Program2/Fedorovsky_summary.pdf Дата изменения: Thu Oct 16 10:30:20 2014 Дата индексирования: Sun Apr 10 16:04:31 2016 Кодировка: Поисковые слова: comet

. .

,
(). , Lp , Lip ­ , .. n-1 P (z ) = k=0 z k Pk (z ), Pk , . 1970-1980 , C (X ) . , . , . . ', . . . , , . ( ) . , . . , . . , . . , . , , , . , . (, Lp Lip ) ( , ) C. . , . ( H 2 ), , K = H 2 H 2 .
c
. . , 2014


Konstantin Fedorovskiy

Approximation by p olyanalytic p olynomials, univalent functions in mo del spaces, and related problems
Summary The proposed pro ject consists of two related parts (topics). The first topic is related with problems of uniform, Lp , and Lip ­ approximation of functions by n-1 polyanalytic polynomials, i.e. by polynomials of the form P (z ) = k=0 z k Pk (z ), where Pk are polynomials in the complex variables, on compact subsets of the complex plane. These problems have appeared in 1970-1980, when first results about density of rational modules in the space C (X ) were obtained by J. Verdera, J. Carmona, A. G. O'Farrell, T. Trent and J. L.-M. Wang and others. As a natural extension of the classical problems of approximation of functions by rational functions and polynomials in the complex variable, this topic has been developed up to the theory of approximation by solutions of elliptic differential equations, which is actively studied at present. Thereupon let us mention (apart the works by the mentioned above authors) the results by A. Boivin, P. M. Gauthier, M. Ya. Mazalov, P. V. Paramonov and other authors, including the author of the proposed pro ject. In spite of the active development of this theory, the problems of polynomial approximation remain unsolved in the general case. The most progress is attained namely in the problem of polyanalytic polynomial approximation, and this progress is peculiarly related with ideas and results of the author of this proposal and his coauthors. In the frameworks of the proposed pro ject it is planned to obtain new necessary and sufficient conditions for approximation (uniform, Lp , and Lip ) of functions by polyanalytic polynomials (of the general form, as well as with certain restrictions to admissible degrees of the conjugate variable) on compact sets in C. The second topic of the pro ject is related with studies of the concept of a Nevanlinna domain and its refinements. These concepts are appeared as the special analytic characteristics of sets in terms of which the solutions of approximation problems under consideration are formulated. By means of the important property of pseudocontinuation of holomorphic functions, these concepts connect the topic under consideration with the theory of model spaces (invariant with respect to the backward shift operator subspaces of the Hardy space H 2 ). In this theory the problem on description of such inner functions that the respective model space K = H 2 H 2 contains bounded univalent functions, and the problem of studying of boundary behavior of such functions are arisen.

c

Konstantin Fedorovskiy,

2014