Finite Gap Jacobi Matrices, II.?The Szeg? Class

Let \frake ì \mathbbR\frak{e}\subset\mathbb{R} be a finite union of disjoint closed intervals. We study measures whose essential support is \frake{\frak{e}} and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to

Finite Gap Jacobi Matrices, I. The Isospectral Torus

Let frake ì mathbbRfrak{e}subsetmathbb{R} be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is  frakefrak{e} , a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.     

Scattering Theory for CMV Matrices: Uniqueness,Helson–Szegő and Strong Szegő Theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.     

On the Szegő Metric

We introduce a new biholomorphically invariant metric based on Fefferman’s invariant Szeg? kernel and investigate the relation of the new metric to the Bergman and Carathéodory metrics. A key tool is a new absolutely invariant function assembled from the Szeg? and Bergman kernels.     

A survey of the Szegő equation

Science China Mathematics - We survey the main properties of the cubic Szegő equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of...     

A Direct Connection Between the Bergman and Szegő Projections

We use Stokes’s theorem to establish an explicit and concrete connection between the Bergman and Szeg? projections on the disc, the ball, and on strongly pseudoconvex domains.     

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

Functional Analysis and Its Applications - We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates...     

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

A mapping connected with the Schur–Szegő composition

Every monic polynomial in one variable of the form $(x+1)S$, $\mathrm{deg}S=n?1$, is presentable in a unique way as a Schur–Szeg? composition of $n?1$ polynomials of the form ${(x+1)}^{n?1}(x+a_i)$. We prove geometric properties of the affine mapping associating to the coefficients of S the $(n?1)$-tuple of values of the elementary symmetric functions of the numbers $a_i$. To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

An Explicit Formula for Szegő Kernels on the Heisenberg Group

In this paper, we give an explicit formula for the Szegö kernel for (0, q) forms on the Heisenberg group H_n+1.     

A new algebraic approach to the Szegő-Widom limit theorem

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if are certain matrix valued functions defined on the unit circle, then

Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

Let be the linear space of the Laurent polynomials and suppose that <, < is a positive-definite Hermitian inner product in with the additional property that . Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <, <, we derive Laurent–Jacobi matrices and for the multiplication operator and its inverse in . These matrices are real and symmetric, and generates a symmetric operator in the Hilbert space ² with natural basis { e _n } _{n = 0} . We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in ² has simple spectrum with generating vector e ₀. Let E be the spectral measure of A. Then the measure _{e 0} given by _{e 0}() =<E() e ₀, e ₀< for all Borel sets in , satisfies forf,g. In this way, we obtain a solution _{e 0} of the Strong Hamburger Moment Problem (SHMP) for which is dense in L ²( _{e 0}). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).     

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.     

Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

Bound states and the Szeg? condition for Jacobi matrices and Schrödinger operators

For Jacobi matrices with a_n=1+(−1)ⁿαn^−γ, b_n=(−1)ⁿβn^−γ, we study bound states and the Szeg? condition. We provide a new proof of Nevai's result that if , the Szeg? condition holds, which works also if one replaces (−1)ⁿ by . We show that if α=0, β≠0, and , the Szeg? condition fails. We also show that if γ=1, α and β are small enough ( will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).     

A Remark on Subgroup Separability in the Class of Finite π-Groups

We prove that if a group G is residually , then for every -subgroup of the group G, the set of -roots from this subgroup is a -separable -subgroup.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).     

Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and such that