Degree of approximation of functions in the Hölder metric by \((e,c)\) means
type: Item
instance of: scholarly article
title: Degree of approximation of functions in the Hölder metric by \((e,c)\) means
zbMATH Open document ID: 0855.42002
DOI: 10.1007/BF02837199
author: Yanyan Li
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1996-09-24T00:00:00Z
review text: Let \(C_{2 \pi}\) denote the Banach space of all \(2 \pi\)-periodic and continuous functions defined on \([- \pi, \pi]\) and let \(H_\alpha\) \((0 < \alpha \leq 1)\) be a subspace of \(C_{2 \pi}\) such that for \(f \in H_\alpha\), \(|f(x) - f(y) |\leq K |x - y |^\alpha\), where \(K > 0\) is some constant. The space \(H_\alpha\) is a Banach space under the norm generated by the Hölder metric [\textit{S. Prössdorf}, Math. Nachr. 69, 7-14 (1975; Zbl 0319.42002)]. The present reviewer [J. Indian Math. Soc., New. Ser. 53, No. 1-4, 99-114 (1988; Zbl 0705.42002) and J. Math. Anal. Appl. 149, No. 1, 236-248 (1990; Zbl 0753.42002)] studied the degree of approximation of functions in Hölder metric by Euler means and Borel means. In this paper, the authors have obtained two theorems concerning the degree of approximation of functions by the \((e,c)\)-mean of its Fourier series in the Hölder metric. One of the theorems reads: Let \(0 < \alpha \leq 1\) and \(0 \leq \beta < \alpha\). Suppose \(f \in H_\alpha\), then \[ \bigl |e_n (f) - f \bigr |_\beta = O(1) \begin{cases} n^{\beta - \alpha} \log n \quad & (0 < \alpha - \beta \leq {1 \over 2}) \\ n^{- {1 \over 2}} \quad & ({1 \over 2} < \alpha - \beta \leq 1) \end{cases}, \] where \(e_n (f;x)\) denotes the \((e,c)\)-mean of \(s_n (f;x)\), the \(n\)th partial sum of the Fourier series of \(f \in H_\alpha\) at \(x\). The other theorem has a different estimate for \(0 < \alpha - \beta \leq {1 \over 2}\) under a Lebesgue-Gergen type condition on \(f\).
reviewed by: Prem Chandra
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 903367
zbMATH Keywords: \((e,c)\)-mean
zbMATH Keywords: Banach space
zbMATH Keywords: Hölder metric
zbMATH Keywords: degree of approximation
zbMATH Keywords: Fourier series
MaRDI profile type: Publication
cites work: Q3485407
cites work: Degree of approximation of functions in the Hölder metric by Borel's means
cites work: Notes on the Theory of Series (XXII)* : On the Tauberian Theorem for Borel Summability
cites work: Q5788558
cites work: Zur Konvergenz der Fourierreihen hölderstetiger Funktionen
cites work: A criterion for the (e, c)-summability of Fourier series
cites work: Q3254471
full work available at URL: https://doi.org/10.1007/bf02837199
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description: scientific article; zbMATH DE number 903367