Cesàro summability and Lebesgue points of higher dimensional Fourier series

Ferenc Weisz

doi:10.3934/mfc.2021033

Article Contents

2022, Volume 5, Issue 3: 241-257. Doi: 10.3934/mfc.2021033

This issue Previous Article Korovkin-type approximation of set-valued and vector-valued functions Next Article New approximation properties of the Bernstein max-min operators and Bernstein max-product operators

Cesàro summability and Lebesgue points of higher dimensional Fourier series

Ferenc Weisz^,

Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary

Corresponding author: Ferenc Weisz
Corresponding author: Ferenc Weisz

Received: May 31, 2021

Revised: September 30, 2021

Early access: November 2021

Published: August 2022

This research was supported by the Hungarian Scientific Research Funds (OTKA) No KH130426

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Abstract

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 $ \sigma_n^{\alpha}f $ of the Fourier series of a multi-dimensional function converge to $ f $ at each Lebesgue point as $ n\to \infty $.

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Mathematics Subject Classification: Primary: 42B08; Secondary 42A24, 42B25.

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Figure 1. Regions of the $ \ell_q $-partial sums for $ d = 2 $

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Figure 2. The cone for $ d = 2 $

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