I would like to rewrite the sum
$\sum_{i=1}^K \sum_{l=-\infty}^\infty \sum_{j=-\infty}^\infty f(i+lK;j-l)$ In the form
$ \dots\sum_{s=-\infty}^\infty \sum_{w=-\infty}^\infty f(s,w)$ where $s=i+lK$, $w=j-l $. How do I do it?
I would like to rewrite the sum
$\sum_{i=1}^K \sum_{l=-\infty}^\infty \sum_{j=-\infty}^\infty f(i+lK;j-l)$ In the form
$ \dots\sum_{s=-\infty}^\infty \sum_{w=-\infty}^\infty f(s,w)$ where $s=i+lK$, $w=j-l $. How do I do it?
I think that the sum is exactly $\sum_{s=-\infty}^\infty \sum_{w=-\infty}^\infty f(s,w)$ Indeed, consider an ordered pair $(s,w)\in\mathbb Z\times\mathbb Z$. The integer $s\in \mathbb Z$ has a unique representation $s=i+lK$ for some $i\in\{1,\dots,K\}$ and $l\in\mathbb Z$. Since $l$ is already determined, from $w=j-l$ we get $j=w+l$. Thus, $(s,w)$ appears in the original sum exactly once.