The Wiener algebra of absolutely convergent Fourier integrals: an overview
type: Item
instance of: scholarly article
title: The Wiener algebra of absolutely convergent Fourier integrals: an overview
zbMATH Open document ID: 1258.42008
published in: Analysis and Mathematical Physics
publication date: 2012-12-04T00:00:00Z
review text: In this nice survey of 68 pages written by three experts in the area, the main results in the topic of representation of non-periodic functions by absolutely convergent Fourier integral are collected, classified, discussed and applied. The history of the topic has its origin in the classical results of S. N. Bernstein and of A. Zygmund, who proved that if a periodic function \(f\) is from the Lip\(\alpha\) class with \(\alpha >1/2\), then for the absolutely convergence of its Fourier series, a sufficient condition is \(\int_{0}^{1}\frac{\omega(f, t)}{t^{3/2}}dt <+\infty\) (S. N. Bernstein's result), and if in addition \(f\) is of bounded variation then a sufficient condition is \(\int_{0}^{1}\frac{\sqrt{\omega(f, t)}}{t}d t < +\infty\) (A. Zygmund's result). Here \(\omega(f, t)\) denotes the modulus of continuity of \(f\). Extensions to the non-periodic case were started in the \(20-40\)s of the twentieth century, with E. C. Titchmarsh's and N. Wiener's results, then becoming a well-studied area of research until our days. This survey consists of 13 sections and an updated list of references. Sections 1, 2 and 3 contains the Introduction, Notation and Definitions and Preliminaries, respectively. Section 4 presents general properties of the Wiener algebras. Sufficient conditions for a function \(f:\mathbb{R}\to \mathbb{C}\) to belong to the Wiener algebra \(W_{0}(\mathbb{R})=\{f ; f(x)=\int_{\mathbb{R}}\mathrm e^{-\mathrm i x y}g(y)d y , g\in L_{1}(\mathbb{R}^{n})\}\) are presented in the next sections 6-12. From a very long list, from each section we selected the following results. Theorem 1. (Tests in one-dimensional case) (i) Let \(f\in L_{p}(\mathbb{R})\bigcap C_{0}(\mathbb{R})\), \(1<p\leq 2\). If \(\int_{\mathbb{R}}|f(x+h)-f(x)|^{p}dx\leq Ch^{\alpha p}\), for all \(0<h\leq 1\), where \(1/p < \alpha \leq 1\), then \(f\in W_{0}(\mathbb{R})\); (ii) If \(f\) satisfies the condition \(|f(x)-f(y)|\leq C\frac{|x-y|^{\lambda}}{(1+|x|)^{\lambda}(1+|y|)^{\lambda}}\), for…
reviewed by: Sorin G. Gal
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 42-02
zbMATH DE Number: 6112460
zbMATH Keywords: Fourier transform
zbMATH Keywords: Fourier multiplier
zbMATH Keywords: radial function
zbMATH Keywords: Bessel function
zbMATH Keywords: fractional derivative
zbMATH Keywords: positive definite function
zbMATH Keywords: bounded variation
zbMATH Keywords: quasi-convexity
author: Elijah Liflyand
author: S. G. Samko
author: R. M. Trigub
MaRDI profile type: Publication
full work available at URL: s13324-012-0025-6
OpenAlex ID: W2056420885
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The Wiener algebra of absolutely convergent Fourier integrals: an overview (DOI)
DOI: 10.1007/S13324-012-0025-6