ORTHOGONAL RATIONAL FUNCTIONS AND NESTED DISKS

In Akhiezer's book [''The Classical Moment Problem and Some Related Qu estions in Analysis,'' Oliver & Boyd, Edinburgh/London, 1965] the uniq ueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plans. T he purpose of the present paper is to develop a similar nested disk th eory for a moment problem that arises in the study of certain orthogon al rational functions. Let {alpha(n)}(n=0)(infinity) be a sequence in thc open unit disk in the complex plant, let B-0 = 1 and B-n (Z) = (k = 0)/Pi(n) \alpha(k)\/<(alpha(k))over bar> 1 - <(alpha(k))over bar>z/a lpha(k) - z,- n = 1, 2, ..., (<(alpha(k))over bar>/\alpha(k)\ = -1 whe n alpha(k) = 0), and let L = span {B-n: n = 0, 1, 2, ...}. We consider the following ''moment'' problem: Given a positive-definite Hermitian inner product (.,.) on L x L, find a non-decreasing function mu on [- pi, pi] (or a positive Borel measure mu on [-pi, pi)) such that [f,g] = integral(-n)(n) f(e(i0)) <(g(e(i0)))over bar> d mu (0) for f, g is a n element of L. In particular we give necessary and sufficient conditi ons for the uniqueness of the solution in the case that (n = 1)Sigma(i nfinity) (1 - \alpha(n)\) < infinity. If this series diverges the solu tion is always unique. (C) 1997 Academic Press.