On the general solution of a functional equation on $\mathbb{Z} \oplus \mathbb{Z}$

Abstract.

Our main goal is to determine the solution of the functional equation $f(x+t, y+t) + f(x-t, y) + f(x, y-t) = f(x-t, y-t) + f(x, y+t) + f(x+t, y)$ where f is a complex valued function defined on the abelian group $\mathbb{Z} \oplus \mathbb{Z}$. This functional equation is connected to a problem in spatial filtering of digital images and also arises while characterizing quadratic polynomials in two variables.