A new class of numerical sequences and its applications to uniform convergence of sine series

TL;DR

This paper introduces a new class of sequences called GM(b,r) and establishes necessary and sufficient conditions for the uniform convergence of sine series with these coefficients, extending previous work by Tikhonov.

Contribution

It generalizes the class of sequences considered for sine series convergence and provides new criteria for uniform convergence based on these sequences.

Findings

Defined the GM(b,r) sequence class.

Derived necessary and sufficient conditions for uniform convergence.

Extended Tikhonov's previous results.

Abstract

In the present paper we introduce a new class of sequences called GM(b,r), which is the generalization of a class considered by Tikhonov. Moreover, we obtained in this note sufficient and necessary conditions for uniform convergence of sine series with (b,r)-general monotone coefficients.