ca.analysis and odes - cauchy product for general case

It's not a problem to multiply the series: the product is $sum_{(t,k)inmathbb Z^2} a_tb_k$. The question is how to sum the double series that we have.

For series with nonnegative terms summation is not a problem either: we take the supremum of all finite sums. And since any finite sum is contained in a sufficiently large square, it follows that $sum_{(t,k)inmathbb Z^2} |a_tb_k|$ is finite whenever $sum_{tinmathbb Z} |a_t|$ and $sum_{kinmathbb Z} |b_k|$ are.

In general, $sum_{(t,k)inmathbb Z^2} a_tb_k=S$ if for any $epsilon>0$ there is a finite subset $Asubset mathbb Z^2$ such that $|sum_{(t,k)in B}a_tb_k - S|<epsilon$ whenever $B$ is finite and $Bsupset A$. Now if both given series converge absolutely, then the contribution from outside of a large square is small, and it follows that $S$ is the product of two sums.