Orthogonal Matrix Polynomials Research Articles

Matrix Jacobi biorthogonal polynomials via Riemann–Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann–Hilbert problem we can derive first and second order differential-difference relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-P I V _{IV} equations are obtained for the three term recurrence relation coefficients.

On the Hahn's property for matrix orthogonal polynomials on the unit circle

In this contribution, we characterize the sequences of matrix orthogonal polynomials on the unit circle (MOPUC) whose derivatives are also orthogonal, i.e. the orthogonal sequences satisfying the Hahn's property. Equivalently, we characterize classical matrix linear functionals on the unit circle.

Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem

We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval [a, b] and their associated second kind polynomials.

Correction to: Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Correction to: Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

Quantum Markov chains on the line: matrix orthogonal polynomials, spectral measures and their statistics

Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. We also consider the case of non-symmetric weight matrices and explore some examples.

Coherent pairs and Sobolev-type orthogonal polynomials on the real line: An extension to the matrix case

In this contribution, we extend the concept of coherent pair for two quasi-definite matrix linear functionals u0 and u1. Necessary and sufficient conditions for these functionals to constitute a coherent pair are determined, when one of them satisfies a matrix Pearson-type equation. Moreover, we deduce algebraic properties of the matrix orthogonal polynomials associated with the Sobolev-type inner product〈p,q〉s=〈p,q〉u0+〈p′M1,q′M2〉u1, where M1 and M2 are m×m non-singular matrices and p,q are matrix polynomials.

Spectral measure of empirical autocovariance matrices of high-dimensional Gaussian stationary processes

Consider the empirical autocovariance matrices at given non-zero time lags, based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measure in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of a shifted version of the autocovariance matrix is obtained. This is then used to obtain the asymptotic behavior of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.

On two resolvent matrices of the truncated Hausdorff matrix moment problem

We consider the truncated Hausdorff matrix moment problem (THMM) in case of a finite number of even moments to be called non degenerate if two block Hankel matrices constructed via the moments are both positive definite matrices. The set of solutions of the THMM problem in case of a finite number of even moments is given with the help of the block matrices of the so-called resolvent matrix. The resolvent matrix of the THMM problem in the non degenerate case for matrix moments of dimension $q\times q$, is a $2q\times 2q$ matrix polynomial constructed via the given moments. In 2001, in [Yu.M. Dyukarev, A.E. Choque Rivero, Power moment problem on compact intervals, Mat. Sb.-2001. -69(1-2). -P.175-187], the resolvent matrix $V^{(2n+1)}$ for the mentioned THMM problem was proposed for the first time. In 2006, in [A. E. Choque Rivero, Y. M. Dyukarev, B. Fritzsche and B. Kirstein, A truncated matricial moment problem on a finite interval, Interpolation, Schur Functions and Moment Problems. Oper. Theory: Adv. Appl. -2006. - 165. - P. 121-173], another resolvent matrix $U^{(2n+1)}$ for the same problem was given. In this paper, we prove that there is an explicit relation between these two resolvent matrices of the form $V^{(2n+1)}=A U^{(2n+1)}B$, where $A$ and $B$ are constant matrices. We also focus on the following difference: For the definition of the resolvent matrix $V^{(2n+1)}$, one requires an additional condition when compared with the resolvent matrix $U^{(2n+1)}$ which only requires that two block Hankel matrices be positive definite. In 2015, in [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259], a representation of the resolvent matrix of 2006 via matrix orthogonal polynomials was given. In this work, we do not relate the resolvent matrix $V^{(2n+1)}$ with the results of [A. E. Choque Rivero, From the Potapov to the Krein-Nudel'man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem, Bol. Soc. Mat. Mexicana. -- 2015. -- 21(2). -- P. 233--259]. The importance of the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$ is explained by the fact that new relations among orthogonal matrix polynomials, Blaschke-Potapov factors, Dyukarev-Stieltjes parameters, and matrix continued fraction can be found. Although in the present work algebraic identities are used, to prove the relation between $U^{(2n+1)}$ and $V^{(2n+1)}$, the analytic justification of both resolvent matrices relies on the V.P. Potapov method. This approach was successfully developed in a number of works concerning interpolation matrix problems in the Nevanlinna class of functions and matrix moment problems.

Non-commutative Hermite–Padé approximation and integrability

We introduce and solve the non-commutative version of the Hermite-Pad\'{e} type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev--Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Pad\'{e} approximants and matrix orthogonal polynomials.

Abelianization of Matrix Orthogonal Polynomials

Abstract The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Padé approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semi-canonical bundle (half-differentials). Several examples are offered in illustration of the new notions.

Deficiency indices of block Jacobi matrices and Miura transformation

Abstract We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

Abstract We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle

Abstract In this contribution, we study properties of block Hessenberg matrices associated with matrix orthonormal polynomials on the unit circle. We also consider the Uvarov and Christoffel spectral matrix transformations of the orthogonality measure, and obtain some relations between the associated Hessenberg matrices.

Discrete Self-adjoint Dirac Systems: Asymptotic Relations, Weyl Functions and Toeplitz Matrices

We consider discrete Dirac systems as an approach to the study of the corresponding block Toeplitz matrices, which in many ways completes the famous approach via Szegő recurrences and matrix orthogonal polynomials. We prove an analog of the Christoffel–Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl–Titchmarsh functions of discrete Dirac systems). These asymptotic relations are expressed also in terms of block Toeplitz matrices. We study the case of rational Weyl–Titchmarsh functions (and GBDT version of the Backlund–Darboux transformation of the trivial discrete Dirac system) as well. It is shown that block diagonal plus block semi-separable Toeplitz matrices (which are easily inverted) appear in this case.

Three-term recurrence relation coefficients and continued fractions related to orthogonal matrix polynomials on the finite interval [a, b

ABSTRACT The four families of matrix orthogonal polynomials are considered arising in the truncated Hausdorff matrix moment (THMM) problem. Two of those families are associated with an odd number of moments and the other two with an even number of moments. The three-term recurrence relations associated with these four families are investigated. Certain explicit formulas are presented relating the three-term recurrence relation coefficients to the Dyukarev-Stieltjes parameters, the Schur complements and the orthogonal matrix polynomials associated with the THMM problem. The matrix version of the J-fraction is presented for the corresponding four extremal solutions of the THMM problem.

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work.

Matrix moment perturbations and the inverse Szegő matrix transformation

Given a perturbation of a matrix measure supported on the unit circle, we analyze the perturbation obtained on the corresponding matrix measure on the real line, when both measures are related through the Szeg˝o matrix transformation. Moreover, using the connection formulas for the corresponding sequences of matrix orthogonal polynomials, we deduce several properties such as relations between the corresponding norms. We illustrate the obtained results with an example.

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

AbstractMatrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

A general family of matrix valued Hermite type orthogonal polynomials is introduced as the matrix orthogonal polynomials with respect to a weight. The matrix polynomials are eigenfunctions of a matrix differential equation. For the weight we derive Pearson equations, which allow us to derive many explicit properties of these matrix polynomials. In particular, the matrix polynomials are eigenfunctions to another matrix differential equation. We also obtain for these polynomials shift operators, a Rodrigues formula, explicit expressions for the squared norm, explicit three term recurrence relations, etc. The matrix entries of the matrix polynomials can be expressed in terms of scalar Hermite and dual Hahn polynomials. We also derive a connection formula for the matrix Hermite polynomials. Next we show that operational Burchnall formulas extend to matrix polynomials. We make this explicit for the matrix Hermite polynomials and for previously introduced matrix Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix Hermite weight. In particular, we obtain an explicit non-trivial solution to the non-abelian Toda lattice equations.