On a New Asymptotic Problem in the Scattering Setting

In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting. The goal of this work is to consider a new asymptotic problem; the related problems in the spectral setting were solved in [4], [6]. With a given Szegö contractive function R on the unite circle T |R(t)| ≤ 1, log(1− |R|) ∈ L (0.1) and a positive measure ν supported on the Blaschke set Z = {ζk : ∑ (1− |ζk|) < ∞}, ν(ζk) = νk > 0, (0.2) we associate the scalar product 〈Df, f〉 = 〈(I − ΓΓ)f, f〉+ ∑ |f(ζk)| νk. (0.3) Here 〈·, ·〉 is the inner standard product in L on T with respect to the Lebesgue measure, Γ is the Hankel operator with the symbol R, Γf = ΓRf = P−(Rf), acting from the Hardy space H into its orthogonal complement H − = L⊖H, P− is the ortho–projection onto H − . Let us point out that we even do not require that the measure ν is finite, thus the scalar product, correspondingly the unbounded operator D, are defined initially on the H–functions that equal zero at all points of Z except for a finite number of them. (In this place we use the Blaschke condition (0.2)). Our generalization deals with the presence of the measure ν. Without it the scalar product plays the key role in the classical now solution of the Nehari problem by Adamyan, Arov and Krein, [1], [2]. For the new point of view on this subject see [9]. The Nehari problem is also known as the generalized Schur problem. Concerning relation of the Schur problem with the Theory of Orthogonal Polynomials on the Unit Circle, CMV matrices and so on, see [7], [8]. For shortness we denote the collection of data by α := {R, ν} Date: February 2, 2008. Partially supported by NSF grant DMS-0200713 and the Austrian Science Found FWF, project number: P16390–N04 . 1 2 and then use the notation H(α) for the closure of admissible functions f from H with respect to the metric (0.3) with D = D(α). The condition (0.1) guaranties that the point evaluation functional (for all ζ0 in the unit disk D) f 7→ f(ζ0) is bounded in H(α). Let k be the reproducing kernel of this space with respect to the origin and let K = k ‖kα‖ . It is almost evident that the system {en(ζ;α)}, en(ζ;α) := ζ Kn(ζ), (0.4) where αn = {ζ R(ζ), |ζ|ν(ζ)} forms an orthonormal basis in H(α). We claim that asymptotically this system behaves as the standard basis system in H, in particulary, that Kn(0) → 1, n → ∞. (0.5) We follow the line of proof that was suggested in [5] and then improved in [10] and [3]. Actually, the general idea is very simple. There are two natural steps in approximation of the given data by “regular” ones. First, to substitute the given measure ν by a finitely supported ν . Second, to substitute R by ρR with 0 < ρ < 1. Then the corresponding data α produce the metric D(α) which is equivalent to the standard metric in H and it is a fairly easy task to prove (0.5) for such data. Further, due to D(α ) ≤ D(α) ≤ D(α) we have the evident estimations K N (0) ≥ K(0) ≥ K ρ (0). And the key point is a certain duality principle, see Corollary 1.6, that will allow us to use the left or right side estimation whenever it is convenient for us. 1. The duality 1.1. The space L(α). We define the outer function Te by |Te| 2 = 1− |R|, Te(0) > 0. Consider the scalar product