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

Resumen

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
$$
(A^{r,\delta}x)_{mn}={\sigma^{r,\delta}_{mn}(x)}=\frac{1}{(m+1)(n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(1+r^j)(1+\delta^k)x_{jk}, \ \ \ \ \ 0<r, \delta<1
$$
The $A^{r,*}$ and $A^{*,\delta}$ transformations are defined respectively by
$$
(A^{r,*}x)_{mn}={\sigma^{r,*}_{mn}(x)}=\frac{1}{m+1}\sum_{j=0}^{m}(1+r^{j})x_{jn}, \ \ \ 0<r<1,
$$
and
$$
(A^{*,\delta}x)_{mn}={\sigma^{*,\delta}_{mn}(x)}=\frac{1}{n+1}\sum_{k=0}^{n}(1+\delta^{k})x_{mk},\ \ \ 0<\delta<1.
$$

We say that $(x_{mn})$ is ($A^{r,\delta}$,1,1) summable to $l$ if $({\sigma^{r,\delta}_{mn}}(x))$ has a finite limit $l$. It is known that if $\lim_{m,n \to \infty }x_{mn}=l$ and $(x_{mn})$ is bounded, then the limit $\lim _{m,n \to \infty} \sigma_{mn}^{r,\delta}(x)=l$ exists.
But the inverse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for ($A^{r,\delta}$,1,1) summability method under which the inverse of this implication holds. Following Tauberian theorems for $(A^{r,\delta},1,1)$ summability method, we also introduce $A^{r,*}$ and $A^{*,\delta}$ transformations of double sequences and obtain Tauberian theorems for the $(A^{r,*},1,0)$ and $(A^{*,\delta},0,1)$ summability methods.