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

Let $G$ be a locally compact group, and let $K$ be a compact subgroup of $G$. Let $\mu : G\longrightarrow\mathbb{C}\backslash\{0\}$ be a character of $G$. In this paper, we deal with the integral equations $$W_{\mu}(K):\; \;\int_{K}f(xkyk^{-1})dk+\mu(y)\int_{K}f(xky^{-1}k^{-1})dk=2f(x)g(y),$$ and $$D_{\mu}(K):\; \;\int_{K}f(xkyk^{-1})dk+\mu(y)\int_{K}f(xky^{-1}k^{-1})dk=2f(x)f(y)$$ for all $x, y\in G$ where $f, g: G\longrightarrow \mathbb{C}$, 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 $f(xy)+\mu(y)f(xy^{-1})=2f(x)f(y)$, recently studied by Davison [18] and Stetk{\ae}r [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 $C_{K}f=\int_{K}f(kxk^{-1})d\omega_{K}(k)\neq 0$ 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 $G=\mathbb{R}$.