On boundedly divergent Walsh-Fourier series
type: Item
instance of: scholarly article
title: On boundedly divergent Walsh-Fourier series
zbMATH Open document ID: 0729.42017
DOI: 10.1007/BF01903853
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: Let \(f^*(x)\) represent the dyadic maximal function of an \(f\in L^ 1[0,1]\) and for any continuous increasing function \(\Phi: [0,\infty)\to [0,\infty)\) let \(H\Phi (H)\) denote the set of functions in \(L^ 1[0,1]\) which satisfy \(\int^{1}_{0}f^*(x)\Phi (f^*(x))dx<\infty.\) The author proves that if \(\Phi (t)=o(\log \log t)\) as \(t\to \infty\) then there is an \(F\in H\Phi (H)\) such that the Walsh-Fourier series of F diverges boundedly a.e. and diverges everywhere on \([0,1].\) The method of proof is elegant and direct, using Waterman's characterization of strongly multiplicative systems to estimate the Walsh-Kolmogorov polynomials and a stopping time like argument to decompose the interval \([0,1]\).
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 4205094
zbMATH Keywords: bounded divergence
zbMATH Keywords: dyadic Hardy spaces
zbMATH Keywords: dyadic maximal function
zbMATH Keywords: Walsh- Fourier series
zbMATH Keywords: Walsh-Kolmogorov polynomials
author: Ferenc Schipp
reviewed by: William R. Wade
MaRDI profile type: Publication
cites work: Q3278441
cites work: Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l'espace L²(0,1)
cites work: Q3207540
cites work: On the Walsh Functions
cites work: Sets of Divergence for Fourier Series
cites work: Almost everywhere convergence of Walsh Fourier series of $H^{1}$-functions
cites work: Q5580436
cites work: Q5583245
cites work: On limits of sequences of operators
cites work: Q5565283
cites work: On systems of functions resembling the Walsh system
description: scientific article; zbMATH DE number 4205094