A generalization of Gajda's equation on commutative topological groups

Abstract

In the present paper we deal with the following generalization of the sine-cosine equation ż f 1 px`y´tq`f 2 px´y`tqdµptq " gpxqhpyq for complex valued functions f 1 , f 2 , g and h defined on a commutative topological group G, where µ is a complex measure defined on G.

On a quadratic type functional equation on locally compact abelian groups

“…Proof. Assume that the functions f, g ∈ C(G) satisfy the functional equation (3). It is easy to check that if we put y = 0 in (3), we get…”

Kannappan’s functional equation on semigroups with involution

“…This is connected to (4), because f is an even solution of (5) if and only if f satisfies (3) (in which R is replaced by G). Recent developments in the theory of Gajda's equation can be found in Fechner and Székelyhidi [6]. Van Vleck's functional equation…”

“…Its solutions are generalized sine functions, while those of (4) are generalized cosine functions. (6) has been studied by Van Vleck [19,20], Perkins and Sahoo [13,Section 3] and Stetkaer [17]. We are of course not the first ones to consider trigonometric functional equations on semigroups.…”

On Gajda’s type quadratic equation on a locally compact abelian group

“…Substituting y by −y in (2.1), and then adding, resp. subtracting the new equation to, 10 resp. from (2.1) we obtain (2.2), resp.…”

“…which was studied by Fechner and Székelyhidi in [8][9][10]. 12 Let µ ∈ M C (G) such that µ(G) :=  G dµ(t) = 1 2 .…”