Integrability of the majorants of Fourier sums and the divergence of Fourier series of functions with restrictions on the integral modulus of continuity.
type: Item
instance of: scholarly article
title: Integrability of the majorants of Fourier sums and the divergence of Fourier series of functions with restrictions on the integral modulus of continuity.
zbMATH Open document ID: 1072.42003
author: N. Yu. Antonov
published in: Mathematical Notes
publication date: 2005-06-21T00:00:00Z
review text: The author proves three essential theorems. The first one asserts that there exists a function from the class \(H_1^{\omega^*},\) where \[ \omega^*(t):=\{\log\log(t^{-1})/\log(t^{-1})\}^{1/2}, \] whose trigonometric Fourier series is divergent almost everywhere. The Theorems 2 and 3 state that if \[ \sum^\infty_{k=1}{\omega(1/k)\over k\,\lambda_k}\leqno(*) \] converges and \(\lambda_k\) satisfies certain conditions, then for any \(f\in H_1^\omega\) \[ \sup_{n\in{\mathbb N}}{| S_n(f,x)| \over\lambda_n} \] is integrable, and \((*)\) is also necessary.
reviewed by: László Leindler
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A20
zbMATH DE Number: 2177673
zbMATH Keywords: Lebesgue integral
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Fourier series
zbMATH Keywords: Hardy estimate
zbMATH Keywords: Dirichlet kernel
zbMATH Keywords: Lagrange finite-increment formula
MaRDI profile type: Publication
full work available at URL: b:matn.0000049660.29081.bc
OpenAlex ID: W2013504758
DOI: 10.1023/B:MATN.0000049660.29081.BC
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description: scientific article; zbMATH DE number 2177673