Extremal solutions of the strong Stieltjes moment problem

A solution of the strong Stieltjes moment problem for the sequence {Cn: n = o,±1, ±2,…} is a finite positive measure μ on [0, ∞) such that cn=∫0∞ tndμ(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 cn=∫−∞∞ tndμ(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 μt corresponding to the constant functions ϕ(z) ≡ t, t ϵ R ∪ {∞}. The accumulation points of the (isolated) set Zt of zeros of β(z)t − δ(z) consists of 0 and ∞. The support of the extremal solution μt is the set Zt ∪ {0}. There exists an interval [t(0), t(∞)] such that the extremal solutions of the Stieltjes problem are exactly those μt for which t ∈ [t(0), t(∞)]. The measures μt(0) and μt(∞) are natural solutions, and the only ones. If ξk(n) denote the zeros of the orthogonal Laurent polynomials determined by {Cn} ordered by size, then {ξk(n)} tends to 0 and {ξn−k(n)} tends to ∞ for arbitrary constant k when n tends to ∞.