On local property of absolute summability of factored Fourier series

TL;DR

This paper develops new theorems on the local properties of absolute summability of factored Fourier series using a generalized summability method, enhancing classical approaches.

Contribution

It introduces two general theorems employing a recently defined absolute summability and a new class, broadening the scope of classical summability results.

Findings

Generalized theorems for local properties of Fourier series

Application of new summability methods to classical results

Improved understanding of absolute summability techniques

Abstract

We establish two general theorems on the local properties of the absolute summability of factored Fourier series by applying a recently defined absolute summability, $\left\vert A,\alpha_{n}\right\vert _{k}$ summability, and the class $\mathcal{S}\left( \alpha_{n},\phi_{n}\right) $, which generalize some well known results and can be applied to improve many classical absolute summability methods.