A necessary condition for the extension of Szegő's asymptotics inside the disk in the Sobolev case

摘要

In the present paper we consider a Sobolev inner product of the following type:〈f(z),g(z)〉s=∫02πf(eiθ)g(eiθ)dμ0(θ)+∫02πf′(eiθ)g′(eiθ)dμ1(θ),z=eiθwith μ0 a finite positive Borel measure on [0,2π] and μ1 a measure in the Szegő class.We assume that the monic Sobolev orthogonal polynomial sequence {φ̃n} satisfies thatlimn→∞φ̃n(z)zn=Π1(1z)Π1(0)uniformly on subsets K of the complex plane such that infz∈K|z|>ρ, where ρ<1 and Π1(z) is the Szegő function of μ1. Then we prove that the sequence of moments of measure μ0, {cn}, satisfies that limn→∞cn=0, and therefore μ0 is a continuous measure.