An indeterminate rational moment problem and Carathéodory functions

摘要

Let {αn}n=1∞ be a sequence of points in the open unit disk in the complex plane and letB0=1andBn(z)=∏k=0nαk¯|αk|αk-z1-αk¯z,n=1,2,…,(αk¯/|αk|=-1 when αk=0). We put L=span{Bn:n=0,1,2,…} and we consider the following “moment” problem:Given a positive-definite Hermitian inner product 〈·,·〉 in L, find all positive Borel measures ν on [-π,π) such that〈f,g〉=∫-ππf(eiθ)g(eiθ)¯dν(θ)forf,g∈L.We assume that this moment problem is indeterminate. Under some additional condition on the αn we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞.