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

TL;DR

This paper investigates integral functional equations on locally compact groups, linking solutions of generalized d'Alembert's and Wilson's equations, and provides explicit solutions using representation theory, with applications to superstability and spherical functions.

Contribution

It introduces a new connection between solutions of integral equations generalizing classical functional equations, and derives explicit solutions using harmonic analysis and representation theory.

Findings

Solutions are characterized via K-spherical functions and irreducible representations.

A link between solutions of Wμ(K) and Dμ(K) is established.

Explicit formulas generalize Euler's formula on groups.

Abstract

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 $G=\mathbb{R}$.