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On a linear functional equation

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In this paper functional equations of the form

$$f(x + ny) + \sum\limits_{k = 0}^{n - 1} {c_k (y)f(x + ky) = 0} $$

are investigated on locally compact Abelian groups. The main result is that, ifG is either a finitely generated discrete Abelian group, or a compactly generated locally compact Abelian group in which the set of compact elements is connected, then all continuous solutionsf of this equation are exponential polynomials. A characterization theorem for exponential polynomials is also proved.

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Székelyhidi, L. On a linear functional equation. Aeq. Math. 38, 113–122 (1989).

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