On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations

Abstract

Let $G$ be a locally compact group, and let $K$ be a compact subgroup of $G$. Let $\mu :G⟶\mathbb{C}\backslash \{0\}$ be a character of $G$. In this paper, we deal with the integral equations $$W_{\mu }(K):{\int }_{K}f(xky{k}^{-1})dk+\mu (y){\int }_{K}f(xk{y}^{-1}{k}^{-1})dk=2f(x)g(y),$$ and $$D_{\mu }(K):{\int }_{K}f(xky{k}^{-1})dk+\mu (y){\int }_{K}f(xk{y}^{-1}{k}^{-1})dk=2f(x)f(y)$$ for all $x,y\in G$ where $f,g:G⟶\mathbb{C}$, to be determined, are complex continuous functions on $G$. When $K\subset Z(G)$, the center of $G$, $D_{\mu }(K)$ reduces to the new version of d'Almbert's functional equation $f(xy)+\mu (y)f(x{y}^{-1})=2f(x)f(y)$, recently studied by Davison [18] and Stetkær [35]. We derive the following link between the solutions of $W_{\mu }(K)$ and $D_{\mu }(K)$ in the following way : If $(f,g)$ is a solution of equation $W_{\mu }(K)$ such that $C_{K}f={\int }_{K}f(kx{k}^{-1})d{\omega }_K(k)\ne 0$ then $g$ is a solution of $D_{\mu }(K)$. This result is used to establish the superstability problem of $W_{\mu }(K)$. In the case where $(G,K)$ is a central pair, we show that the solutions are expressed by means of $K$-spherical functions and related functions. Also we give explicit formulas of solutions of $D_{\mu }(K)$ in terms of irreducible representations of $G$. These formulas generalize Euler's formula $\cos (x)=\frac{e^{ix}+e^{-ix}}{2}$ on $G=\mathbb{R}$.

19 pages