Bultheel, Adhemar × González-Vera, Pablo Hendriksen, Erik Njåstad, Olav #

Issue Date:

May-2002

Publisher:

Elsevier

Series Title:

Applied mathematics and computation vol:128 issue:2-3 pages:217-235

Abstract:

Let {a_k} be a sequence of points on the real axis, and let L denote the linear span of the set {1/w_0,1/w_1,...1/w_n,...} where w_0=1, w_n = (z-a_1)... (z-a_n) for n = 1,2,... A measure µ with support in an interval [g,∞) and with respect to which all functions in L.L are integrable determines an inner product in L and hence orthogonal rational functions {f_n} corresponding to the sequence {1/w_n} and associated functions {s_n}. Assuming certain constellations on the points {a_k}, it is shown that the sequences {-s_{2m}/f_{2m}} and {-s_{2m+1}/f_{2m+1}} are monotonic on a certain real interval (a,b) and convergent in the complex plane outside the interval [g,∞) to functions F(z,µ^{(0)}) and F(z,µ^{(∞)}), where F(z,µ) denotes the Stieltjes transform
∫_{x=-∞...∞} dµ(t)/(t-x)
of a measure µ. Furthermore for every µ giving rise to the same integral values for the functions in L.L, the inequality
F(x,µ^{(0)}) ≤ F(x,µ) ≤ F(x,µ^{(∞)})
holds for x in (a,b). 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.