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

Abstract

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma :S\to S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) +\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) -\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] 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.

Acta Mathematica Hungarica