Wilson’s functional equation for vector and matrix functions

Sinopoulos, Pavlos

doi:10.1090/S0002-9939-97-03685-X

Wilson’s functional equation for vector and matrix functions
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Proc. Amer. Math. Soc. 125 (1997), 1089-1094 Request permission

Abstract:

We determine the general solution of the functional equation \[ f(x+y)+f(x-y) =A(y)f(x)\qquad (x,y\in G), \] where $G$ is a 2-divisible abelian group, $f$ is a vector-valued function and $A$ is a matrix-valued function. Using this result we solve the scalar equation \[ f(x+y)+f(x-y)=g_1(x)h_1(y)+g_2(x) h_2(y)\qquad (x,y\in G), \] which contains as special cases, among others, the d’Alembert and Wilson equations and the parallelogram law.

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