Peter Yuditskii

Asymptotic behavior of polynomials orthonormal on a homogeneous set

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 Szegö-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 Szegö; 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 Schrödinger equations.

Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points

Let a be a positive measure whose support is an interval E plus a denumerable set of mass points which accumulate at the boundary points of E only. Under the assumptions that the mass points satisfy Blaschkes condition and that the absolutely continuous part of a satisfies Szegos condition, asymptotics for the orthonormal polynomials on and off the support are given. So far asymptotics were only available if the set of mass points is finite.

An abstract interpolation problem and the extension theory of isometric operators

The algebraic structure of V.P. Potapov’s Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov-Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type.

An Analysis and Extension of V.P. Potapov’s Approach to Interpolation Problems with Applications to the Generalized Bi-Tangential Schur-Nevanlinna-Pick Problem and J-Inner-Outer Factorization

There exist several approaches to the generalization and unification of various problems of the Nevanlinna-Pick type [1–4, 5, 9, 10, 12, 14, 28, 29, 31, 35, 38, 40].

ON GENERALIZED SUM RULES FOR JACOBI MATRICES

This work is in a stream (see e.g. (4), (8), (10), (11), (7)) initiated by a paper of Killip and Simon (9), an earlier paper (5) also should be mentioned here. Using methods of Functional Analysis and the classical Szego Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.

On a conjecture of Widom

In 1969 Harold Widom published his seminal paper (Widom, 1969) which gave a complete description of orthogonal and Chebyshev polynomials on a system of smooth Jordan curves. When there were Jordan arcs present the theory of orthogonal polynomials turned out to be just the same, but for Chebyshev polynomials Widoms approach proved only an upper estimate, which he conjectured to be the correct asymptotic behavior. In this note we make some clarifications which will show that the situation is more complicated.

Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces

We give an explicit parametrization of a set of almost periodic CMV matrices whose spectrum (is equal to the absolute continuous spectrum and) is a homogenous set E lying on the unit circle, for instance a Cantor set of positive Lebesgue measure. First to every operator of this set we associate a function from a certain subclass of the Schur functions. Then it is shown that such a function can be represented by reproducing kernels of appropriated Hardy spaces and, consequently, it gives rise to a CMV matrix of the set under consideration. If E is a finite system of arcs our results become basically the results of Geronimo and Johnson.

Full length article: On the L 1 extremal problem for entire functions

We generalize the Korkin–Zolotarev theorem to the case of entire functions having the smallest L1 norm on a system of intervals E . If C \ E is a domain of Widom type with the Direct Cauchy Theorem, we give an explicit formula for the minimal deviation. Important relations between the problem and the theory of canonical systems with reflectionless resolvent functions are shown. c ⃝ 2013 Elsevier Inc. All rights reserved.

On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples Peter Yuditskii

We discuss several questions which remained open in our joint work with M. Sodin “Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of characterautomorphic functions”. In particular, we show that there exists a non-homogeneous set E such that the Direct Cauchy Theorem (DCT) holds in the Widom domain CE. On the other hand we demonstrate that the weak homogeneity condition on E (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.

Finite Difference Operators with a Finite-Band Spectrum

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so-called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the 2d+1-diagonal matrix; the d-root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to the particular covering, which generate almost periodic CMV matrices recently widely discussed. Then we study an important theme: covering of one spectral surface by another one and the related transformations on the set of multidiagonal operators (so-called Renormalization Equations). We prove several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and for the case of a rational double covering.