Approximation of analytic functions by polynomials with integer coefficients
type: Item
instance of: scholarly article
title: Approximation of analytic functions by polynomials with integer coefficients
zbMATH Open document ID: 0882.41003
DOI: 10.1007/BF02310951
published in: Mathematical Notes
publication date: 1997-07-14T00:00:00Z
review text: In this paper, the author investigates the degree (or rate) of approximation by polynomials with integer coefficients for complex valued functions \(f\) defined on the unit square \(G: =[0,1] \times [0,1]\). Let \(A(G)\) denote the class of functions continuous on \(G\) and analytic in its interior. For \(r\geq 1\), let \(A^r(G)\) denote the class of functions \(f\), such that \(f^{(j)} \in A(G)\), \(0\leq j\leq r\). Let \(\Gamma\) denote the boundary of \(G\) and \(\{z_j\}^4_{j=1}\) denote its four corners. Let \(f\in A^r(G)\) and let \(q\) be the interpolating polynomial determined by the conditions. \[ q^{(s)}(z_j)= f^{(s)}(z_j),\;1\leq j\leq 4,\;0\leq s\leq r. \] Assume that the coefficients of \(q\) are integers (this is the major integer assumption). Then there exists an algebraic polynomial \(Q\) of degree \(\leq n\) with coefficients in \(\mathbb{Z}[i]\) and such that of \(z\in\Gamma\) and \(0\leq s \leq r\), \[ |f^{(s)}- Q^{(s)} |(z) \leq M \rho^{r-s}_{1+1/n} (z) \omega \bigl(f^{(s)}; \rho_{1+1/n} (z) \bigr). \] Here \(M\) is independent of \(z,n\); \(\rho_{1+1/n} (z)\) is the distance from \(z\) to the level curve \(\Gamma_{1+1/n}\), and \(\omega (f^{(s)}; \cdot)\) denotes the modulus of continuity of \(f^{(s)}\) on \(G\). The level curve of course refers to a level curve of the conformal mapping of the exterior of \(G\) onto the exterior of the unit ball. The author relates this to results of Belyi on the degree of approximation of functions on a domain whose boundary is quasiconformal, and to work of Trigub and Alper on polynomials with integer coefficients, especially on the real line.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1028496
zbMATH Keywords: integer polynomials
zbMATH Keywords: complex approximation
zbMATH Keywords: degree of approximation
author: Valeriĭ V. Volchkov
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
cites work: Q3937799
cites work: Q3884383
cites work: CONFORMAL MAPPINGS AND THE APPROXIMATION OF ANALYTIC FUNCTIONS IN DOMAINS WITH A QUASICONFORMAL BOUNDARY
cites work: Q5522339
cites work: Q3882893
cites work: Q5577334
full work available at URL: https://doi.org/10.1007/bf02310951
OpenAlex ID: W1981730445
description: scientific article; zbMATH DE number 1028496