International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 16, Pages 847-859
doi:10.1155/S0161171204304254
On Cauchy-type functional equations
1Department of Mathematics, Faculty of Sciences, University of Ibnou Zohr, Agadir 80000, Morocco
2Department of Mathematics, Faculty of Sciences, Semlalia, University of Cadi Ayyad, Marrakech 40000, Morocco
Received 24 April 2003
Copyright © 2004 Elqorachi Elhoucien and Mohamed Akkouchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let G be a Hausdorff topological locally compact group. Let M(G) denote the Banach algebra of all complex and bounded measures on G. For all integers n≥1 and all μ∈M(G), we consider the functional equations ∫Gf(xty)dμ(t)=∑i=1ngi(x)hi(y), x,y∈G, where the functions f, {gi}, {hi}: G→ℂ to be determined are bounded and continuous functions on G. We show how the solutions of these equations are closely related to the solutions of the μ-spherical matrix functions. When G is a compact group and μ is a Gelfand measure, we give the set of continuous solutions of these equations.