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 $G=\mathbb{ R}$.
Categories: math.CA math.FA