Extremal solutions of the strong Stieltjes moment problem

Abstract

A solution of the strong Stieltjes moment problem for the sequence { C n : n = o,±1, ±2,…} is a finite positive measure μ on [0, ∞) such that c n= ∫ 0 ∞ t ndμ(t) for all n, while a solution of the strong Hamburger moment problem for the same sequence is a finite positive measure μ on (−∞, ∞) such that c n= ∫ −∞ ∞ t ndμ(t) for all n. When the Hamburger problem is indeterminate, there exists a one-to-one correspondence between all solutions μ and all Nevanlinna functions ϕ, the constant ∞ included. The correspondence is given by Fμ(z)=− ∝(z)ϕ(z)−γ(z) β(z)ϕ(z)−δ(z) , where α, β, γ, δ are certain functions holomorphic in C − {0} . The extremal solutions are the solutions μ t corresponding to the constant functions ϕ( z) ≡ t, t ϵ R ∪ {∞} . The accumulation points of the (isolated) set Z t of zeros of β( z) t − δ( z) consists of 0 and ∞. The support of the extremal solution μ t is the set Z t ∪ {0}. There exists an interval [ t (0), t (∞)] such that the extremal solutions of the Stieltjes problem are exactly those μ t for which t ∈ [ t (0), t (∞)]. The measures μ t (0) and μ t (∞) are natural solutions, and the only ones. If ξ k ( n) denote the zeros of the orthogonal Laurent polynomials determined by { C n } ordered by size, then { ξ k ( n) } tends to 0 and { ξ n− k ( n) } tends to ∞ for arbitrary constant k when n tends to ∞.