Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups

TL;DR

This paper characterizes solutions to specific Wilson-type functional equations on semigroups involving an involutive automorphism and measures supported on the center, expanding understanding of these equations in algebraic structures.

Contribution

It provides a comprehensive description of solutions to Kannappan-Wilson and Van Vleck-Wilson equations on semigroups with measures supported on the center, a novel extension in the field.

Findings

Solutions are characterized explicitly in terms of the semigroup structure.

The equations are solved for measures that are linear combinations of Dirac measures at central elements.

The results have implications for the symmetry properties of solutions in algebraic contexts.

Abstract

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma:S\rightarrow S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)d\mu(t) +\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)d\mu(t) -\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in S,\] where $\mu$ is a measure that is a linear combination of Dirac measures $(\delta_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented.