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
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
We also show that every function \({g: G \rightarrow P}\) , fulfilling (A), is a solution of (B) if and only if A = G.
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Bahyrycz, A., Brzdȩk, J. On solutions of the d’Alembert equation on a restricted domain. Aequat. Math. 85, 169–183 (2013). https://doi.org/10.1007/s00010-012-0162-x
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DOI: https://doi.org/10.1007/s00010-012-0162-x