I was working in a problem in number theory and I blocked over the problem:
Given functions $f:\mathbb{Z}\to \mathbb{Z}$, $g:\mathbb{Z^2}\to \mathbb{Z}$ and $h:\mathbb{Z^2}\to \mathbb{Z}$ such that $ \forall m,n \in \mathbb{Z}$: $$ f(m+n)=g(m,n)+h(m,n) $$ $$ f(m-n)=g(m,n)-h(m,n) $$ can we prove that $g$ depends only on $m$ and $h$ depends only on $n$.
I'm working in a problem similar but more complicated, for which are interested, given a quadratic form with two variables $Q(m,n)$, I'm looking for a characterization of functions $f$ such that $f\big(Q(m,n)\big)$ can be written as $Q(m',n')$ for all integers $m,n$.
No, for example $f(n)=n^2$ gives $g(m,n)=m^2+n^2$. There are many more examples, since what you're asking is essentially that $f(m+n)+f(m-n)$ is a function of $m$ (taking even values only) and that $f(m+n)-f(m-n)$ is a function of $n$ (taking even values). These requirements are clearly not satisfied in general.