Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Wilson's functional equation for vector and matrix functions

Author(s): Pavlos Sinopoulos
Journal: Proc. Amer. Math. Soc. 125 (1997), 1089-1094.
MSC (1991): Primary 39B42, 39B52, 39B62
MathSciNet review: 1363186
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Abstract: We determine the general solution of the functional equation

$\begin{displaymath}f(x+y)+f(x-y) =A(y)f(x)\qquad (x,y\in G), \end{displaymath}$

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

$\begin{displaymath}f(x+y)+f(x-y)=g_1(x)h_1(y)+g_2(x) h_2(y)\qquad (x,y\in G), \end{displaymath}$

which contains as special cases, among others, the d'Alembert and Wilson equations and the parallelogram law.

References:

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