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 | References | Similar Articles | Additional Information

Abstract: We determine the general solution of the functional equation

where is a 2-divisible abelian group, is a vector-valued function and is a matrix-valued function. Using this result we solve the scalar equation

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

**1.**J. Aczél, J. K. Chung and C. T. Ng,*Symmetric second differences in product form on groups*, Topics in mathematical analysis, (edited by Th. M. Rassias), World Scientific Publ., 1989, pp. 1-22. MR**92g:39007****2.**J. Aczél and J. Dhombres,*Functional equations in several variables*, Cambridge Univ. Press, 1989. MR**90h:39001****3.**J. K. Chung, Pl. Kannappan and C. T. Ng,*On two trigonometric functional equations*, Math. Rep. Toyama Univ.**11**(1988), 153-165. MR**89j:39010****4.**A. L. Rukhin,*The solution of the functional equation of d'Alembert's type for commutative groups*, Intern. J. Math. Sci.**5**(1982), 315-335. MR**84g:39006****5.**P. Sinopoulos,*Generalized sine equations*, I, Aeq. Mathematicae**48**(1994), 171-193. MR**95i:39020****6.**P. Sinopoulos,*Generalized sine equations*, II, Aeq. Mathematicae**49**(1995), 122-152. MR**96b:39020****7.**P. Sinopoulos,*A functional equation in three variables for five unknown functions*, Submitted.**8.**D. A. Suprunenco and R. I. Tyshkevich,*Commutative matrices*, Academic Press, 1968.**9.**W. H. Wilson,*On certain related functional equations*, Bull. Amer. Math. Soc.**26**(1919-20), 300-312.

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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