Unique solvability of an extended Hamburger moment problem

Abstract

R-functions are rational functions with no poles in the extended complex plane outside a given set { a 1,…, a p } of points on the real axis. Methods from the theory of orthogonal polynomials can be extended to R-functions. By this means the author solved an extended Hamburger moment problem: Given sequences of real numbers { c n ( i) : n = 1, 2,…}, i = 1,…, p, find conditions for a distribution function ψ to exist such that ∫ −α α dψ(t) = 1, ∫ −α α dψ(t) (t − a i) m = c m (i) m = 1, 2,…, i = 1,…, p . In this paper these methods are extended to treat conditions for the moment problem to have a unique solution. The results are related to the classical limit point-limit circle situation.