On local property of absolute summability of factored Fourier series

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, ${\left|A,{\alpha }_n\right|}_k$ summability, and the class $\mathcal{S}\left({\alpha }_n,{\varphi }_n\right)$, which generalize some well known results and can be applied to improve many classical absolute summability methods.