Asymptotic behavior of polynomials orthonormal on a homogeneous set

Abstract

Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $\sigma$ be a positive measure with $\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous part of $\sigma$ satisfies a Szeg\"o--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $\sigma$. For the special case that $E$ consists of a finite number of intervals and that $\sigma$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szeg\"o and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schr\"odinger equations.