Wiener-Ingham type inequality for Vilenkin groups and its application to harmonic analysis
type: Item
instance of: scholarly article
title: Wiener-Ingham type inequality for Vilenkin groups and its application to harmonic analysis
zbMATH Open document ID: 1374.43002
author: Bhikha Lila Ghodadra
author: J. R. Patadia
published in: Periodica Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
review text: An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x) = \sum A_k e^{in_k x}\), where the \(A_k\)'s are complex numbers, \(n_{-k} = -n_k\) and \(\{n_k \}\) satisfies the small gap condition \((n_{k+1} - n_k) \geq q \geq 1\) for \(k = 0, 1, 2,\ldots\), says that if \(I\) is any subinterval of \([-\pi, \pi]\) of length \(| I| = 2\pi(1 +\delta)/q > 2\pi/q\) then \(\sum | A_k | ^2 \leq A_{\delta} | I| ^{-1}\int_{I} | f | ^2, | A_k | \leq A_{\delta} | I| ^{-1}\int_{I} | f | \), wherein \(A_{\delta}\) depends only on \(\delta\). In their paper the authors prove such an inequality in the setting of the Vilenkin groups \(G\). The inequality is then applied to generalize the Bernstěin, Szász and Stečhkin type results concerning the absolute convergence of Fourier series on \(G\).
reviewed by: G. Gát
Mathematics Subject Classification ID: 43A75
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6699249
zbMATH Keywords: absolute convergence
zbMATH Keywords: Wiener-Ingham inequality
zbMATH Keywords: Vilenkin group
zbMATH Keywords: Fourier series
zbMATH Keywords: small gaps
MaRDI profile type: Publication
full work available at URL: s10998-015-0095-y
OpenAlex ID: W1901085783
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DOI: 10.1007/S10998-015-0095-Y
description: scientific article; zbMATH DE number 6699249