𝐵𝑉-functions on commutative semigroups

Abstract

The concept of functions of bounded variation on a linearly ordered set has been generalized to a distributive lattice [1] and more recently to a semilattice (cf. [3], [4] and [5]). Here, using different techniques, we further extend this notion to commutative semigroups with identity and show that the BK-functions characterize the "abstract moment sequences" or what we call moment functions.Let S be a commutative semigroup with identity 1. A nontrivial homomorphism, which maps S into the multiplicative semigroup of nonnegative real numbers not greater than 1, will be called an exponential We will denote the set of all exponentials on S by exp(S). Equipped with the topology of simple convergence, exp(S) is a compact Hausdorff space. We now formulate the abstract moment problem. Given a real-valued function ƒ on 5, when does there exist a regular Borel measure \x s on exp(S) such that f(x) = j eeexp ( S ) e (