Triangular Cesàro summability of two dimensional Fourier series
type: Item
instance of: scholarly article
title: Triangular Cesàro summability of two dimensional Fourier series
zbMATH Open document ID: 1240.42029
author: Ferenc Weisz
published in: Acta Mathematica Hungarica
publication date: 2011-12-19T00:00:00Z
review text: The paper is devoted to the triangular summability of Fourier series, more precisely, the author shows that the maximal operator of the triangular \((C,\alpha)\) Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space \(H_p(\mathbb T^2)\) to \(L_p(\mathbb T^2)\) for all \(p\) from the range \(2/(2+\alpha)<p\leq\infty\) and, consequently, is of weak type \((1,1)\). The proof is very much two-dimensional, and it is not clear what may happen in higher dimension. As a consequence, an old Herriot's result (proved in a different way) follows as the triangular Cesàro means converge almost everywhere. It is worth mentioning that triangular summability has a longer history than that of which the reader may get an impression from the paper. We give only two absolutely necessary additional references: \textit{O. I. Kuznetsova} [Sib.\ Mat.\ Zh.\ 18, 629--636 (1977; Zbl 0364.42018)] and \textit{A. N. Podkorytov} [Vestn.\ Leningr.\ Univ., Math.\ 11, 197--205 (1983; Zbl 0517.42020)].
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 5990617
zbMATH Keywords: Hardy space
zbMATH Keywords: \(p\)-atom
zbMATH Keywords: interpolation
zbMATH Keywords: Fourier series
zbMATH Keywords: triangular summation
zbMATH Keywords: Cesàro summability
MaRDI profile type: Publication
full work available at URL: s10474-011-0095-1
OpenAlex ID: W2042177998
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DOI: 10.1007/S10474-011-0095-1
description: scientific article; zbMATH DE number 5990617