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What is a general solution of the equation $$ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0 \textrm{ for } x,y \in \mathbb R, h>0, $$ with unknown function $f: \mathbb R^2 \rightarrow \mathbb R$?

Functions of the form $f(x,y)=g(x)+h(y)$ are solutions. Are there another ones?

Thanks

Michael Hardy
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Richard
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1 Answers1

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If $a(x)$ and $b(x)$ are periodic with period $h$, then another solution is \begin{align} f(x,y) = a(x)b(y). \end{align}

Doubt
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    Seems that your solution can be generalized taking any double-periodic function, i.e. invariant under all shifts in $(h\vec{e_1}, h\vec{e_2})$ lattice, $\vec{e_1} = (1, 0)$, $\vec{e_2} = (0, 1)$. – Evgeny Jul 30 '13 at 04:17