Integral functional equations on locally compact groups with involution

Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group G that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s and Van Vleck’s equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group G. So the theory is part of harmonic analysis on groups.