Absolute convergence of series of Fourier coefficients with respect to an orthonormal system
type: Item
instance of: scholarly article
title: Absolute convergence of series of Fourier coefficients with respect to an orthonormal system
zbMATH Open document ID: 0780.42018
author: Tran Minh
published in: Moscow University Mathematics Bulletin
publication date: 1993-03-11T00:00:00Z
review text: The paper under review is connected with the absolute convergence of the Fourier coefficients (or that of a subsequence) in terms of the modulus of continuity. S. B. Stechkin proved that for any subsequence \(\{m_ k\}\) of the natural numbers one has \[ \sum_ k(| a_{m_ k} |+| b_{m_ k} |) \leq C\sum{1\over \sqrt k} \omega^{(2)} \left({1 \over m_ k};f \right), \] where \(\omega^{(2)}\) is the \(L^ 2\)-modulus of continuity of \(f\). The author found in an earlier paper that the convergence of the left hand side for every function from the smoothness class \(H^ \omega\) is equivalent to the convergence of \(\sum{1\over \sqrt k} \omega \left({1\over m_ k} \right)\). In the present paper he extends these investigations to the absolute convergence of Fourier series with respect to arbitrary orthonormal systems on [0,1]. In particular, he obtains the complete analogue of the aforementioned result for general orthonormal systems under a mild and natural assumption on the modulus of continuity \(\omega\).
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 130688
zbMATH Keywords: absolute convergence
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Fourier series
zbMATH Keywords: orthonormal systems
reviewed by: Vilmos Totik
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 130688