Approximation by entire functions on a countable union of segments on the real axis. III: Further generalization
type: Item
instance of: scholarly article
title: Approximation by entire functions on a countable union of segments on the real axis. III: Further generalization
zbMATH Open document ID: 1405.30036
DOI: 10.3103/S1063454118020085
published in: Vestnik St. Petersburg University. Mathematics
publication date: 2018-10-11T00:00:00Z
review text: This is a continuation of the authors' previous papers [Vestn. St. Petersbg. Univ., Math. 49, No. 4, 373--378 (2016); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2016, No. 4, 644--650 (2016; Zbl 1393.30029); Vestn. St. Petersbg. Univ., Math. 50, No. 1, 35--43 (2017); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 62, No. 1, 53--63 (2017; Zbl 1382.41007)]. The problem is to approximate a smooth function \(f\) defined on a set \(E=\bigcup_{n\in\mathbb{Z}}[a_n,b_n]\), is a countable union of disjunct intervals on the real line, by functions of exponential type \(\sigma\geq 1\). In the previous papers \(f\) was taken from a Hölder class. In this paper the class is a significantly larger class \(\Lambda_M^{r+\omega}\) with \(r\geq 1\), where \(\Lambda_M^\omega\) consists of complex functions such that \(|f(x)|\leq M\), \(x\in E\) and \(|f(x_2)-f(x_1)|\leq c_f\omega(|x_2-x_1|)\) where \(\omega\) is a modulus of continuity satisfying \(\int_0^x \omega(t)/t\, dt +x\int_x^\infty \omega(t)/t^2\,dt\leq c\omega(x)\). As in the previous papers, also the intervals of \(E\) have to satisfy certain conditions but these are not changed. The main theorem proves the existence of an approximant \(F_\sigma\) such that \(|f(x)-F_\sigma(x)|\) is bounded in terms of \(\omega\) and the distance from \(x\) to the Joukowsky map of the closest interval \([a_n,b_n]\) from \(E\).
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 6952192
zbMATH Keywords: smooth functions
zbMATH Keywords: entire functions of exponential type
zbMATH Keywords: approximation on a subset of the real axis
author: O. V. Sil'vanovich
author: N. A. Shirokov
MaRDI profile type: Publication
cites work: Approximations by entire functions on countable unions of segments of the real axis
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full work available at URL: s1063454118020085
OpenAlex ID: W2808340931
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description: scientific article; zbMATH DE number 6952192