On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group

[摘要] Let $G$ be a compact group. Let $sigma$ be a continuous involutionof $G$. In this paper, we areconcerned by the following functional equation$$int_{G}f(xtyt^{-1}),dt+int_{G}f(xtsigma(y)t^{-1}),dt=2g(x)h(y), quadx, y in G,$$ where $f, g, h colonG mapsto mathbb{C}$, to bedetermined, are complex continuous functions on $G$ such that $f$ iscentral. This equation generalizes d'Alembert's and Wilson'sfunctional equations. We show that the solutions are expressed bymeans of characters of irreducible, continuous and unitaryrepresentations of the group $G$.

[学科分类] 数学（综合）

[关键词] Compact groups;Functional equations;Central functions;Lie;groups;Invariant differential operators.