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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Wilson's functional equation
for vector and matrix functions


Author: Pavlos Sinopoulos
Journal: Proc. Amer. Math. Soc. 125 (1997), 1089-1094
MSC (1991): Primary 39B42, 39B52, 39B62
DOI: https://doi.org/10.1090/S0002-9939-97-03685-X
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.


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Additional Information

Pavlos Sinopoulos
Affiliation: 18 Vergovitsas Street, GR-11475 Athens, Greece

DOI: https://doi.org/10.1090/S0002-9939-97-03685-X
Received by editor(s): August 4, 1995
Received by editor(s) in revised form: September 22, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society