A rational Stieltjes moment problem

摘要

Let {αk} be a sequence of points on the real axis, and let L denote the linear span of the set1ω0,1ω1,…,1ωn,…,where ω0=1,ωn=(z−α1)(z−α2)⋯(z−αn)forn⩾1. A measure μ with support in an interval [γ,∞) and with respect to which all functions in L·L are integrable induces an inner product in L and hence orthogonal rational functions {ϕn} and associated functions {σn} corresponding to the sequence {1/ωn}. Assuming certain constellations of the points {αk}, it is shown that the sequences−σ2m(z)ϕ2m(z)and−σ2m+1(z)ϕ2m+1(z)are monotonic on a certain real interval (α,β) and convergent in the complex plane outside the interval [γ,∞) to functions F(z,μ(0)) and F(z,μ(∞)), where F(z,μ) denotes the Stieltjes transform ∫−∞∞dμ(t)/(t−z) of a measure μ. Furthermore for every μ giving rise to the same integral values for functions in L·L, the inequality F(x,μ(0))⩽F(x,μ)⩽F(x,μ(∞)) holds for x∈(α,β). These results are essentially generalizations of results concerning the strong (or two-point) Stieltjes moment problem, and are also similar to results concerning the classical Stieltjes moment problem.