Necessary and sufficient Tauberian conditions under which convergence follows from summability $A^{r, p}$

In this paper, we introduce the summability method $A^{r, p}$ and obtain necessary and sufficient Tauberian conditions under which the ordinary convergence of a sequence follows from its summability $A^{r, p}$. The main results are new Tauberian theorems for the summability method $A^{r, p}$, which are generalizations of the corresponding Tauberian theorems for the summability method $A^r$ introduced by Başar.