A density problem for orthogonal rational 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−αkz,n=1,2,…,(αk/|αk|=−1whenαk=0). We putL=span{Bn:n=0,1,2,…}and we consider the following ‘moment’ problem:Given a positive-definite Hermitian inner product 〈·,·〉 on L×L, find a nondecreasing function μ on [−π,π] (or a positive Borel measure μ on [−π,π)) such that〈f,g〉=∫−ππf(eiθ)g(eiθ)dμ(θ)forf,g∈L.We give a necessary and sufficient condition (called ‘N-extremality’) on a solution μ of the moment problem in order that L is dense in L2μ.