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

Let $x=(x_{mn})$ be a double sequence of real or complex numbers. The $A^{r,\delta}$-transform of a sequence $(x_{mn})$ is defined by $$ (A^{r,\delta}x)_{mn}={\sigma^{r,\delta}_{mn}(x)}=\frac{1}{(m+1)(n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(1+r^j)(1+\delta^k)x_{jk}, \ \ \ \ \ 0