On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions

Abstract

In this paper we g i v e some necessary and su cient conditions for uniform approximability of functions by polyanalytic polynomials on plane compact sets of special form. Also connections with the corresponding Dirichlet problem are considered.

Density of certain polynomial modules

“…The specific analytic nature of approximability conditions in this problem was discovered in [9]. In [6] the approximability criterion for Carathéodory compact sets was obtained as well as in [3,7] several necessary and sufficient conditions on a general compact set X in order that the equality A(X, z) = P(X, z) holds were established (see also [15] and references therein). The present paper is the first in which Problem 1, for the case d > 1, is considered.…”

C m -Approximation by Polyanalytic Polynomials on Compact Subsets of the Complex Plane

“…In [7,Example 4.5] it was shown, that there exists a non Carathéodory compact set Y such that P 2 (Y ) = C(Y ), but P 3 (Y ) = C(Y ). Furthermore, in view of [11,Theorem 1], for each integer k ≥ 1 there exists a non Carathéodory compact set X ⊂ C such that P 2k (X ) = C(X ) = P k (X ).…”

“…Recall, that a compact set X is said to be a Carathéodory compact set if ∂ X = ∂ X , where X denotes the union of X and all bounded connected components of the set C\X . It was proved in [7,Theorem 2.2], that if X is a Carathéodory compact set in C, then A n (X ) = P n (X ) if and only if each bounded connected component of the set C\X is not a Nevanlinna domain (see [7,Definition 2.1] and [6,Definition 3]). The concept of a Nevanlinna domain is a special analytic characteristic of a set.…”

“…In [7] Problem 1.2 was solved for Carathéodory compact sets. Recall, that a compact set X is said to be a Carathéodory compact set if ∂ X = ∂ X , where X denotes the union of X and all bounded connected components of the set C\X .…”

“…In this case Problem 1.2 is more difficult and sophisticated, and still remains unsolved in the general case. One ought to notice that several necessary and sufficient conditions in order that A n (X ) = P n (X ) have been obtained in [5][6][7][8][9].…”