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

On local property of absolute summability of factored Fourier series