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, $\left\vert A,\alpha_{n}\right\vert _{k}$ 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.