Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations

TL;DR

This paper characterizes solutions and stability properties of specific variants of Van Vleck's and d'Alembert's functional equations on semigroups, involving complex measures and involutive morphisms.

Contribution

It provides explicit solutions for these functional equations on semigroups with measures and involutions, and establishes superstability results.

Findings

Explicit solutions for the functional equations on semigroups.

Conditions for continuous solutions involving measures and involutions.

Superstability theorems for the first functional equation.

Abstract

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive morphism of $S$, and $\mu$ is a complex measure that is linear combinations of Dirac measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation $$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(\sigma(y)tx)d\upsilon(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a topological semigroup, $\sigma$ is a continuous involutive automorphism of $S$, and $\upsilon$ is a complex measure with compact support and which is $\sigma$-invariant. (3) We prove the superstability theorems of the first functional equation.