On local property of absolute summability of factored Fourier series

We establish two general theorems on the local properties of the absolute summability of factored Fourier series by applying a recently defined absolute summability, ${\mid A,{\alpha }_n\mid }_k$ summability, and the class $\mathcal{S}({\alpha }_n,{\varphi }_n)$, which generalize some well known results and can be applied to improve many classical absolute summability methods.