An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups
type: Item
instance of: scholarly article
title: An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups
zbMATH Open document ID: 1388.43004
DOI: 10.1134/S2070046617040057
published in: \(p\)-Adic Numbers, Ultrametric Analysis, and Applications
publication date: 2018-04-13T00:00:00Z
review text: Suppose that \(f\in L^2(\mathbb{R})\), \(0<\alpha<1\). Titchmarsh's theorem states that \(f\) satisfies the \(L^2\) Hölder condition \[ \|f(\cdot - h) - f(\cdot)\|_{L^2(\mathbb{R})}=O(h^{\alpha}), \quad h\to 0, \] if and only if \[ \int_{|t|>r}|\widehat{f}(t)|^2\,dt = O(r^{-2\alpha}), \quad r\to\infty, \] where \(\widehat{f}\) is the Fourier transform of \(f\). The author proves an analogue of this theorem for functions on locally compact Vilenkin groups.
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A10
zbMATH DE Number: 6859804
zbMATH Keywords: harmonic analysis on Vilenkin groups
zbMATH Keywords: Titchmarsh theorem
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Fourier transform
zbMATH Keywords: Vilenkin groups
author: Sergei S. Platonov
reviewed by: Yuri A. Farkov
MaRDI profile type: Publication
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full work available at URL: s2070046617040057
OpenAlex ID: W2766643428
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description: scientific article; zbMATH DE number 6859804