Expansion of functions of quasianalytic classes in series of polynomials
type: Item
instance of: scholarly article
title: Expansion of functions of quasianalytic classes in series of polynomials
zbMATH Open document ID: 0715.41045
DOI: 10.1007/BF01060545
author: V. G. Khryptun
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Delta_ n(a)\) be an arbitrary domain of the n-dimensional space of the points \(x=(x_ 1,x_ 2,...,x_ n)\) that is star-shaped with respect to the point \(a=(a_ 1,a_ 2,...,a_ n)\); \(n=1,2,...<+\infty\); \(\delta_ n(a)\) be an arbitrary closed domain contained in \(\Delta_ n(a)\), \(\delta_ n(a)\ni a\); and \(\{\) M(q)\(\}\) \((q=0,1,2,...)\) be an arbitrary sequence of positive numbers. We put an infinitely differentiable function f(x), defined in \(\Delta_ n(a)\), in the class \(C_ n\{M(q)\}\), if in each domain \(\delta_ n(a)\subseteq \Delta_ n(a)\) \[ (1)\quad | D_ 1^{q_ 1}D_ 2^{q_ 2}...D_ n^{q_ n}f(x)| \leq C^{q_ 1+q_ 2+...+q_ n}M(q_ 1)M(q_ 2)....\quad M(q_ n), \] and in the class \(C_{n,1}\{M(q)\}\) if \((2)\quad | d^ qf(x)| \leq C^ qM(q).\) In equalities (1) and (2) the constant C depends on f(x) and \(\delta_ n(a)\); \(D_ i\) is the partial derivative with respect to the i-th argument; \(q_ i\) are nonnegative integers; df(x) is the differential of f(x) at the point x, and the numbers M(q) are the same for all functions of each class. The classes \(C_ n\{M(q)\}\) and \(C_{n,1}\{M(q)\}\) are said to be quasianalytic if the functions that belong to each of the classes are uniquely determined by their values and the values of all their partial derivatives at any point \(b\in \Delta_ n(a)\). Let \(\Delta_ 1(a)\) be a star-shaped (with respect to the point a) domain in the plane of the complex variable x and \(C_ 1\{q!\}\) be the class of the analytic functions f(x) of one variable x in \(\Delta_ 1(a)\). Mittag-Leffler has shown that the functions of the class \(C_ 1\{q!\}\) can be expanded in a uniformly convergent series of polynomials inside the domain \(\Delta_ 1(a):\) \[ (3)\quad f(x)=\sum^{\infty}_{\nu =0}(\sum^{p_{\nu}}_{q=0}C_{q,\nu}f^{(q)}(a)(x-a)^ q), \] where the degrees \(p_{\nu}\) of the polynomials and the numbers \(c_{q,\nu}\) are the same for all functions of the calss \(C_ 1\{q!\}\). In the present article, for the functions that belong to t…
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 4177813
zbMATH Keywords: absolutely convergent series of polynomials
zbMATH Keywords: quasianalytic classes
MaRDI profile type: Publication
cites work: Q4742751
cites work: Q3246558
description: scientific article; zbMATH DE number 4177813