On Ces\'aro summability of Fourier series of functions from multidimensional Waterman classes

TL;DR

This paper extends Waterman's results on Cesàro summability of Fourier series to multidimensional functions, highlighting the importance of continuity in convergence and localization of Cesàro means.

Contribution

It generalizes one-dimensional Cesàro summability results to multidimensional Waterman classes, emphasizing the role of continuity for convergence.

Findings

Continuity of functions is essential for Cesàro convergence in multiple dimensions.

Multidimensional case differs from one-dimensional, requiring additional conditions.

Results extend the understanding of Fourier series summability in higher dimensions.

Abstract

An analogue of D. Waterman's result on the summability of the Fourier series for functions of bounded \Lambda-variation by the Ces\'aro methods of negative order is obtained in multidimensional case. It is proved that, unlike one-dimensional case, the continuity of function in the corresponding variation is essential for the convergence and even for the localization of the Ces\'aro means for certain orders of these means.