scholarly journals Lebesgue points of multi-dimensional functions

2008 ◽  
Vol 21 (3) ◽  
pp. 255-265
Author(s):  
Ferenc Weisz
Keyword(s):  
Lebesgue Point ◽  
Lebesgue Points ◽  
Higher Dimensional ◽  
L Log L

Lebesgue and Walsh-Lebesgue points are introduced for higher dimensional functions and it is proved that a.e. point is a (Walsh)-Lebesgue point of a function f from the space L(log L)d-1. Every function f ? L(log L)d-1 is Fej?r summable at each (Walsh)-Lebesgue point. .

scholarly journals Cesàro summability and Lebesgue points of higher dimensional Fourier series

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Maximal Functions ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
New Concepts ◽  
Higher Dimensional ◽  
Different Types

<p style='text-indent:20px;'>We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means <inline-formula><tex-math id="M1">\begin{document}$ \sigma_n^{\alpha}f $\end{document}</tex-math></inline-formula> of the Fourier series of a multi-dimensional function converge to <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> at each Lebesgue point as <inline-formula><tex-math id="M3">\begin{document}$ n\to \infty $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

scholarly journals Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Fourier Transforms ◽  
Lebesgue Point ◽  
Strong Summability ◽  
Wiener Amalgam Spaces ◽  
Wiener Amalgam Space ◽  
Almost Everywhere ◽  
Lebesgue Points ◽  
Amalgam Spaces

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, iffis in the Wiener amalgam spaceW(L1,lq)(R)andfis almost everywhere locally bounded, orf∈W(Lp,lq)(R)  (1<p<∞,1≤q<∞), then strongθ-summability holds at each Lebesgue point off. The analogous results are given for Fourier series, too.

scholarly journals Generalized Lebesgue Points for Hajłasz Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Toni Heikkinen
Keyword(s):  
Function Space ◽  
Measure Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Lebesgue Point ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Lebesgue Points ◽  
Boyd Index ◽  
Generalized Lebesgue Point

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

scholarly journals Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2724
Author(s):  
Ziwei Li ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Lebesgue Point ◽  
Metric Measure Spaces ◽  
Exceptional Sets ◽  
Lebesgue Points ◽  
Measure Spaces ◽  
Zero Capacity

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to these spaces. In case these functions are not locally integrable, the authors also consider their generalized Lebesgue points defined via the γ-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets.

Lebesgue points and restricted convergence of Fourier transforms and Fourier series

2016 ◽  
Vol 15 (01) ◽  
pp. 107-121 ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Fourier Transforms ◽  
Lebesgue Point ◽  
Summability Method ◽  
Herz Space ◽  
Lebesgue Points ◽  
Special Cases ◽  
General Summability ◽  
De La Vallée Poussin

In this paper, a general summability method of multi-dimensional Fourier transforms, the so-called [Formula: see text]-summability, is investigated. It is shown that if [Formula: see text] is in a Herz space, then the summability means [Formula: see text] of a function [Formula: see text] converge to [Formula: see text] at each modified Lebesgue point, whenever [Formula: see text] and [Formula: see text] is in a cone. The same holds for Fourier series. Some special cases of the [Formula: see text]-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, Cesàro, de la Vallée-Poussin, Rogosinski and Riesz summations.

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