A Density Problem For Orthogonal Rational Functions

Let fff ng 1 n=1 be a sequence of points in the open unit disk in the complex plane and let B 0 = 1 and B n (z) = n Y k=0 ff k jff k j ff k \Gamma z 1 \Gamma ff k z ; n = 1; 2; : : : ; ( ff k jff k j = \Gamma1 when ff k = 0). We put L = spanfB n : n = 0; 1; 2; : : :g and we consider the following "moment" problem: Given a positive-definite Hermitian inner product h\Delta; \Deltai on L \Theta L, find a non-decreasing function ¯ on [\Gammaß; ß] (or a positive Borel measure ¯ on [\Gammaß; ß)) such that hf; gi = Z ß \Gammaß f(e i` )g(e i` )d¯(`) for f; g 2 L: We give a necessary and sufficient condition (called "N-extremality") on a solution ¯ of the moment problem in order that L is dense in L 2 ¯ . KEYWORDS: orthogonal rational functions, moment problem, nested disks, N-extremal. AMS SUBJECT CLASSIFICATION: primary 30E05. 1. Introduction. In [2] it is shown that a solution ¯ of the Hamburger moment problem is N-extremal if and only if the set of polynomials is de..