Generalized localization for spherical partial sums of the multiple Fourier series and integrals

TL;DR

This paper addresses the open problem of generalized localization for spherical partial sums of multiple Fourier series, providing a positive solution and surveying related convergence issues.

Contribution

It offers the first positive solution to the generalized localization problem for spherical partial sums of multiple Fourier series.

Findings

Positive solution for generalized localization of spherical partial sums

Survey of convergence and localization in Fourier analysis

Advances understanding of almost-everywhere convergence

Abstract

It is well known, that Luzin's conjecture has a positive solution for one dimensional trigonometric Fourier series and it is still open for the spherical partial sums $S_\lambda f(x)$, $f\in L_2(\mathbb{T}^N)$, of multiple Fourier series, while it has the solution for square and rectangular partial sums. Historically progress with solving Luzin's conjecture has been made by considering easier problems. One of such easier problems for $S_\lambda f(x)$ was suggested by V. A. Il'in in 1968 and this problem is called the generalized localization principle. In this paper we first give a short survey on convergence almost-everywhere of Fourier series and on generalized localization of Fourier integrals, then present a positive solution for the generalized localization problem for $S_\lambda f(x)$.