# Generalized localization for spherical partial sums of multiple Fourier series

## TL;DR

This paper proves the generalized localization principle for spherical partial sums of multiple Fourier series in the $L_2$ class, showing convergence to zero almost everywhere on sets where the function is zero, and clarifies the limits of this property in other $L_p$ spaces.

## Contribution

It establishes the validity of generalized localization for $L_2$ functions and completes the characterization of this property across all $L_p$ spaces.

## Key findings

- Generalized localization holds for $L_2$ functions.
- Localization fails for $L_p$ with $p<2$.
- The problem is fully solved for all $L_p$ spaces.

## Abstract

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the $L_2$ - class is proved, that is, if $f\in L_2(T^N)$ and $f=0$ on an open set $\Omega \subset T^N$, then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on $\Omega$. It has been previously known that the generalized localization is not valid in $L_p(T^N)$ when $1\leq p<2$. Thus the problem of generalized localization for the spherical partial sums is completely solved in $L_p(T^N)$, $p\geq 1$: if $p\geq2$ then we have the generalized localization and if $p<2$, then the generalized localization fails.