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

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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.

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Received: 19 February 2002; revised manuscript accepted: 14 May 2002

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Sahoo, P., Székelyhidi, L. On the general solution of a functional equation on $\mathbb{Z} \oplus \mathbb{Z}$. Arch. Math. 81, 233–239 (2003).

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