I´ve just started studying double series with index over the integers, and I´ve been traying to prove the next theorem:
If $$\sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}|a_{i,j}| < \infty \Rightarrow \sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}|a_{j,i-j}| < \infty$$
and $$\sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}a_{i,j} = \sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}a_{j,i-j}$$
That is we can rearrange the double series and sum diagonally. I have succesfully proved that $$\sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}|a_{j,i-j}| < \infty$$ but I´m lost with the second part of the proof. Here are my definitions:
Definition 1: A double series over the integers $\sum_{i = -\infty}^{\infty}a_i$ converges ( is absolute convergent) to a real number iff $\sum_{i = 0}^{\infty}a_i$ and $\sum_{i = 1}^{\infty}a_{-i}$ converge separately (are absolute convergent separately) and in this case:
$$\sum_{i = -\infty}^{\infty}a_i = \sum_{i = 0}^{\infty}a_i + \sum_{i = 1}^{\infty}a_{-i}$$
I also proved that under absolute convergence we have the following:
Theorem : $\lim_{n\to\infty}\sum_{i = -n}^{n}|a_i|$ exists iff $\sum_{i = 0}^{\infty}|a_i| , \sum_{i = 1}^{\infty}|a_{-i}|$ both exists (and are finite) and $$\lim_{n\to\infty}\sum_{i = -n}^{n} a_i=\sum_{i = 0}^{\infty}a_i + \sum_{i = 1}^{\infty}a_{-i} = \sum_{i = -\infty}^{\infty}a_i$$
Definition 2: The iterated double series $\sum_{i = -\infty}^{\infty}\sum_{j = -\infty}^{\infty}a_{i,j}$ converges iff :
1) For fixed $i \in \mathbb{Z}$, $\sum_{j = -\infty}^{\infty}a_{i,j}$ converges to a real number $b_i$ and
2)$\sum_{i = -\infty}^{\infty}b_i$ converges to a real number
I have tried to adapt the proof when we have a series of the form: $\sum_{n = 0}^{\infty}a_n$ and under absolute convergesnce we can rearrange it´s terms and the sums are equal, but I was unsuccesfull.
Another questions that I have is the following: How can we define a double sum over the integers (that is not an iterated sum): $$\sum_{i ,j= -\infty}^{\infty}a_{i,j}$$ ?
I would really appreciate any hints or suggestions with this problem.
Thank you.