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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

\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 [Enhancements On Off] (What's this?)

  • 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

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