Summary
The observation that the solutions to d'Alembert's functional equation are Z2-spherical functions onR 2 gives us a natural way of extending d'Alembert's functional equation to groups. We deduce in this setting that the general solutions are joint eigenfunctions for a system of partial differential operators, and we find a formula for the bounded solutions.
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Stetkaer, H. D'Alembert's equation and spherical functions. Aeq. Math. 48, 220–227 (1994). https://doi.org/10.1007/BF01832986
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DOI: https://doi.org/10.1007/BF01832986