Lebesgue points of two-dimensional Fourier transforms and strong summability
type: Item
instance of: scholarly article
title: Lebesgue points of two-dimensional Fourier transforms and strong summability
zbMATH Open document ID: 1333.42008
published in: The Journal of Fourier Analysis and Applications
publication date: 2015-12-03T00:00:00Z
review text: The recent literature of the Fourier analysis contains various studies concerning a general summation method called \(\theta\)-method. This method is generated by a single function \(\theta\) and includes Fejér, Riesz, Weierstrass, Abel and other known summation methods. The author proved in 2002 and 2011 that the Marcinkiewics means generated by the \(\theta\)-summation in the case \(f \in L_1(\mathbb{R})\) converge almost everywere to \(f\). In this paper, this result is generalized to Wiener amalgam spaces \(W(L_p,l_\infty)(\mathbb{R}^2)\), \(1\leq p\leq \infty\). The set of convergence is characterized by using the concepts of modified strong Lebesgue points in the case \(p=1\) and modified Lebesgue points in the case \(1<p<\infty\). Moreover, some classical results for the one-dimensional strong summability and for strong \(\theta\)-summability are generalized. Finally, the paper gives also some applications to various summability methods.
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42A24
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 6515786
zbMATH Keywords: \(\theta\)-summability
zbMATH Keywords: Marcinkiewichz summability
zbMATH Keywords: Fourier transforms
zbMATH Keywords: modified Lebesgue points
zbMATH Keywords: strong summability
author: Ferenc Weisz
reviewed by: Delfina Roux
MaRDI profile type: Publication
full work available at URL: s00041-015-9393-2
OpenAlex ID: W2061696633
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DOI: 10.1007/S00041-015-9393-2
description: scientific article; zbMATH DE number 6515786