Wilson’s functional equation for vector and matrix functions
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- by Pavlos Sinopoulos PDF
- 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.References
- J. Aczél, J. K. Chung, and C. T. Ng, Symmetric second differences in product form on groups, Topics in mathematical analysis, Ser. Pure Math., vol. 11, World Sci. Publ., Teaneck, NJ, 1989, pp. 1–22. MR 1116572, DOI 10.1142/9789814434201_{0}001
- J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. With applications to mathematics, information theory and to the natural and social sciences. MR 1004465, DOI 10.1017/CBO9781139086578
- J. K. Chung, Pl. Kannappan, and C. T. Ng, On two trigonometric functional equations, Math. Rep. Toyama Univ. 11 (1988), 153–165. MR 974257
- A. L. Rukhin, The solution of the functional equation of d’Alembert’s type for commutative groups, Internat. J. Math. Math. Sci. 5 (1982), no. 2, 315–335. MR 655518, DOI 10.1155/S0161171282000301
- Pavlos Sinopoulos, Generalized sine equations. I, Aequationes Math. 48 (1994), no. 2-3, 171–193. MR 1295090, DOI 10.1007/BF01832984
- Pavlos Sinopoulos, Generalized sine equations. II, Aequationes Math. 49 (1995), no. 1-2, 122–152. MR 1309298, DOI 10.1007/BF01827933
- P. Sinopoulos, A functional equation in three variables for five unknown functions, Submitted.
- D. A. Suprunenco and R. I. Tyshkevich, Commutative matrices, Academic Press, 1968.
- W. H. Wilson, On certain related functional equations, Bull. Amer. Math. Soc. 26 (1919–20), 300–312.
Additional Information
- Pavlos Sinopoulos
- Affiliation: 18 Vergovitsas Street, GR-11475 Athens, Greece
- Received by editor(s): August 4, 1995
- Received by editor(s) in revised form: September 22, 1995
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- 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