Orthogonal Rational Functions and Nested Disks Original Research Article

In Akhiezerʹs book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {αn}∞n=0be a sequence in the open unit disk in the complex plane, letB0=1andBn(z)=∏k=0n αk|αk| αk−z1−αkz,n=1, 2, …,(αk/|αk|=−1 whenαk=0), and letL=span{B: n=0, 1, 2, …}.We consider the following “moment” problem: Given a positive-definite Hermitian inner product ⦠·, ·⦔ on L×L, find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that⦠f, g⦔=∫π−π f(eiθ) g(eiθ) dμ(θ) forf, g∈L.In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that∑n=1∞ (1−|αn|)<∞.If this series diverges the solution is always unique.