Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (μ₀,μ₁) is a coherent pair of measures on the unit circle, then μ₀ is a semi-classical measure. Moreover, we obtain that the linear functional associated with μ₁ is a specific rational transformation of the linear functional corresponding to μ₀. Some examples are given.     

Positive Lyapunov exponents for a class of ergodic orthogonal polynomials on the unit circle

Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C¹ function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent.     

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.     

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r_k}_k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r_k}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.     

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials B_n(^.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

A class of vector‐valued dilation‐and‐modulation frames on the half real line

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θ_a transform in , and using Θ_a‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.     

Mass points of measures on the unit circle and reflection coefficients

Measures on the unit circle and orthogonal polynomials are completely determined by their reflection coefficients through the Szego recurrences. We find the conditions on the reflection coefficients which provide the lack of a mass point at . We show that the result is sharp in a sense.     

Ladder operators for Szego polynomials and related biorthogonal rational functions

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

     

Strong Asymptotics for the Continuous Sobolev Orthogonal Polynomials on the Unit Circle

Strong (or Szeg -type) asymptotics for orthogonal polynomials with respect to a Sobolev inner product with general measures (the first measure is arbitrary and the second one is absolutely continuous and satisfying a smoothness condition) is obtained. Examples, illustrating the theorems proved, are presented.     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

Measures on the unit circle and unitary truncations of unitary operators

In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (a_n). Among other cases, we study the derived set of the support of the measure when lim_n|a_n+1/a_n|=1, obtaining a natural generalization of the known result for the López class , lim_n|a_n|(0,1).On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials.Finally, we use the previous results to discuss the domain of convergence of rational approximants of Carathéodory functions, including the convergence on the unit circle.     

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.     

Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems — II

We derive the Christoffel–Geronimus–Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z₁,…,z_M} the bi-orthogonal system is known to be monodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel–Geronimus–Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system — the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota–Miwa equations are derived for the τ-functions or equivalently Toeplitz determinants of the system.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.