On a variant of μ-Wilson’s functional equation on a locally compact group

Abstract

Let G be a locally compact group. Let σ be a continuous involution of G and let μ be a complex bounded and σ-invariant measure. We determine the continuous, bounded and μ-central solutions of the functional equation $$ \int\limits_{G} f(xty)d \mu (t) + \int\limits_{G} f(\sigma (y) tx) d \mu(t) = 2f(x)g(y),\, \quad x,y \in G. $$ The paper of Stetkaer (Aequationes Math 68(3):160–176, 2004) is the essential motivation for this result and the methods used here are closely related to and inspired by it. In addition, when μ is compactly supported, we will investigate the superstability of this functional equation, which is bounded by the unknown functions \({\varphi (x)}\) or \({\varphi (y)}\).