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

Let \(S\) be a semigroup, \(Z(S)\) the center of \(S\) and \(\sigma \colon S \rightarrow S\) is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation

\(\int_{S} f(xyt)\, d\mu(t) + \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\)

and the Van Vleck-Wilson functional equation

\(\int_{S} f(xyt)\, d\mu(t) - \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\)

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.