A new class of matrices and its applications to absolute summability factor theorems

We introduce a new class of matrices $\mathcal{S}(a_n,{\varphi }_n)$ and establish a general absolute summability factor theorem by applying a recently defined absolute summability, ${|A,a_n|}_k$ summability, and the class $\mathcal{S}(a_n,{\varphi }_n)$ . As applications of the main result, we further verify that many well‐known classes of matrices such as Cesàro's matrices, Rhaly’s generalized Cesàro matrices, the $p$ ‐Cesàro matrices, and Riesz matrices are all in $\mathcal{S}(a_n,{\varphi }_n)$ for some certain sequences $\left\{a_n\right\}$ and $\left\{{\varphi }_n\right\}$ , and, therefore, many new factor theorems can be obtained under suitable conditions.