Solutions and stability of a generalization of Wilson's equation

TL;DR

This paper investigates solutions and stability properties of a generalized Wilson's functional equation on locally compact groups, extending classical results to noncommutative settings with measure and involution considerations.

Contribution

It introduces a generalized functional equation involving measures and involutions, and establishes new stability results on noncommutative groups.

Findings

Solutions satisfy a related functional equation when f is non-zero.

Stability theorems are proven for the generalized Wilson's equation.

Results extend classical Wilson's equation to noncommutative groups with measure considerations.

Abstract

In this paper we study the solutions and stability of the generalized Wilson's functional equation $\int_{G}f(xty)d\mu(t)+\int_{G}f(xt\sigma(y))d\mu(t)=2f(x)g(y),\; x,y\in G$, where $G$ is a locally compact group, $\sigma$ is a continuous involution of $G$ and $\mu$ is an idempotent complex measure with compact support and which is $\sigma$-invariant. We show that $\int_{G}g(xty)d\mu(t)+\int_{G}g(xt\sigma(y))d\mu(t)=2g(x)g(y),\; x,y\in G$ if $f\neq 0$ and $\int_{G}f(t.)d\mu(t)\neq 0$. We also study some stability theorems of that equation and we establish the stability on noncommutaive groups of the classical Wilson's functional equation $f(xy)+\chi(y)f(x\sigma(y))=2f(x)g(y)\; x,y\in G$, where $\chi$ is a unitary character of $G$.