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

Let S be a semigroup, Z ( S ) the center of S and σ : S → S is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation ∫ S f ( x y t ) d μ ( t ) + ∫ S f ( σ ( y ) x t ) d μ ( t ) = 2 f ( x ) g ( y ) , x , y ∈ S , and the Van Vleck-Wilson functional equation ∫ S f ( x y t ) d μ ( t ) - ∫ S f ( σ ( y ) x t ) d μ ( t ) = 2 f ( x ) g ( y ) , x , y ∈ S , where μ is a measure that is a linear combination of Dirac measures ( δ z i ) i ∈ I , such that z i ∈ Z ( S ) for all i ∈ I . Interesting consequences of these results are presented.