The "sum of this sort" is not a distribution unless sum is really finite. And in the latter case $A$ is supported on the diagonal $\{(x,y)\colon x=y\}$. So the answer is "No"

The "sum of this sort" is not a distribution unless sum is really finite. And in the latter case $A$ is supported on the diagonal $\{(x,y)\colon x=y\}$. So the answer is "No"

Actually, there are distributions of the infinite order, f.e. \begin{equation} \sum _{n\ge 0} \delta ^{(n)} (x-n). \end{equation} but these $\delta$-functions are located at points tending to the border of the domain (which here is $+\infty$)