, Volume 85, Issue 1-2, pp 169-183,
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On solutions of the d’Alembert equation on a restricted domain

Abstract

Let A be a subgroup of a commutative group (G, + ) and P be a commutative ring. We give the full description of functions ${g: G \rightarrow P}$ satisfying $$g(x + y) + g(x - y) = 2g(x)g(y) \quad (x, y) \in A \times G. \quad\quad\quad\quad (A)$$ Thus we obtain a family of functions depicting evolutions of quite arbitrary functions ${g_0 : G \to P}$ into cosine functions ${g: G \to P}$ , i.e., solutions of the d’Alembert (cosine) functional equation $$g(x + y) + g(x - y) = 2g(x)g(y) \quad x, y \in G. \quad\quad\quad\quad (B)$$ We also show that every function ${g: G \rightarrow P}$ , fulfilling (A), is a solution of (B) if and only if A = G.