On a functional equation of Aczél and Chung

Summary

This paper is concerned mainly with the functional equation

$$\sum\limits_{\imath = 0}^m {F_\imath (\alpha _\imath ,x + \beta _i y)} = \sum\limits_{k = 1}^n {G_k (x)H_k (y)} $$

((1))

which is a generalization of the Levi-Civita equation

$$f(x + y) = \sum\limits_{k = 1}^N {g_k (x)h_k (y).} $$

((2))

For complex valued functions of a real variable, Aczél and Chung [1] have shown that (under certain additional natural assumptions) the locally Lebesgue integrable solutions of (1) are exponential polynomials. Jarai [6] has shown that the local integrability assumption can be weakened to measurability. Our aim is to solve distributional analogues of (1) and (2) and thereby obtain another generalization of the result of Aczél and Chung. Essentially, we will show that (1) can be reduced to (2).