Generalized localization for spherical partial sums of multiple Fourier series

2019/01/10 by Ravshan Ashurov, Ashurov, Ravshan
Mathematics · #42B05 #42B99 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1901.03028

openalex publication_date 2019/01/10 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L₂ - class is proved, that is, if f∈ L₂(T^N) and f=0 on an open set Ω⊂ T^N, then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on Ω. It has been previously known that the generalized localization is not valid in L_p(T^N) when 1≤ p<2. Thus the problem of generalized localization for the spherical partial sums is completely solved in L_p(T^N), p≥ 1: if p≥2 then we have the generalized localization and if p<2, then the generalized localization fails.

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