On a functional equation related to digital filtering

Summary.

Let $f: G \times G \to {\Bbb C}$, where G denotes a 2-divisible abelian group and ${\Bbb C}$ the set of complex numbers. The general solution of the functional equation¶¶f(x + t,y + t) + f(x - t,y) + f(x,y - t) = f(x - t,y - t) + f(x,y + t) + f(x + t,y)¶ for all x,y,t in G is determined. It is shown that the solution of this functional equation is of the form $f(x,y) = B(x,y)+\varphi(x) + \psi(y) + \chi (x-y)$, where the map $B : G \times G \to {\Bbb C}$ is biadditive and $\varphi, \psi, \chi :G\to{\Bbb C}$ are arbitrary functions.