Asymptotic behavior of polynomials orthonormal on a homogeneous set

Abstract

LetE be a homogeneous compact set, for instance a Cantor set of positive length. Further, let σ be a positive measure with supp(σ)=E. Under the condition that the absolutely continuous part of σ satisfies a Szego-type condition, we give an asymptotic representation, on and off the support, for the polynomials orthonomal with respect to σ. For the special case thatE consists of a finite number of intervals and that σ has no singular component, this is a well-known result of Widom. IfE=[a,b], it becomes a classical result due to Szego; 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 ofE which are supposed to accumulate only onE 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. In other words, the Jacobi matrices associated with the above discussed orthonomal polynomials are compact perturbations of a onesided 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 Schrodinger equations.