On the uniform convergence of double sine series

TL;DR

This paper establishes new sufficient and necessary conditions for the uniform convergence of double sine series, generalizing previous results by introducing a new class of coefficient sequences.

Contribution

It introduces a novel class of double sequences and provides new conditions for the uniform convergence of double sine series, extending classical results.

Findings

New sufficient conditions for uniform convergence.

Necessary conditions for non-negative coefficients.

Generalization of coefficient classes for convergence analysis.

Abstract

The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [3], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.