Properties of matrix orthogonal polynomials via their Riemann-Hilbert characterization

@article{Grunbaum2011PropertiesOM,
  title={Properties of matrix orthogonal polynomials via their Riemann-Hilbert characterization},
  author={F. Alberto Grunbaum and Manuel Dom{\'i}nguez de la Iglesia and Andrei Mart{\'i}nez-Finkelshtein},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2011},
  volume={7},
  pages={098}
}
We give a Riemann{Hilbert approach to the theory of matrix orthogonal poly- nomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of lad- der operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a… 

Asymptotics of matrix valued orthogonal polynomials on ~$[-1,1]$

We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert

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

  • C. Charlier
  • Mathematics, Computer Science
    Transactions of Mathematics and Its Applications
  • 2021
It is shown 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.

Ladder relations for a class of matrix valued orthogonal polynomials

Using the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on R , and

Zeros and ratio asymptotics for matrix orthogonal polynomials

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients An and Bn having limits A and B, respectively, (the matrix Nevai class) were obtained by Durán. In the present paper,

Zeros and ratio asymptotics for matrix orthogonal polynomials

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients An and Bn having limits A and B, respectively, (the matrix Nevai class) were obtained by Durán. In the present paper,

The Algebra of Differential Operators for a Gegenbauer Weight Matrix

In this paper we study in detail algebraic properties of the algebra $\mathcal D(W)$ of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order

Abelianization of Matrix Orthogonal Polynomials

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

Matrix-valued orthogonal polynomials related to hexagon tilings

This paper shows that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane and finds the limiting zero distribution of the upper right entry under a geometric condition on the curve Σ̃0 that was unable to prove, but was convincingly supported by numerical evidence.

References

SHOWING 1-10 OF 57 REFERENCES

Orthogonal matrix polynomials satisfying second-order differential equations

We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Abstract We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

We introduce a family of weight matrices W of the form T(t)T*(t), , where is a certain nilpotent matrix and is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size

Szegő's Theorem and Its Descendants: Spectral Theory for L 2 Perturbations of Orthogonal Polynomials

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gabor Szego's classic 1915 theorem and its 1920 extension.

Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory

In the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting. An analogue of the determinant definition of orthogonal polynomials is

Random Matrix Theory: Invariant Ensembles and Universality

This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles - orthogonal, unitary, and symplectic. The authors follow the

Riemann–Hilbert Problems, Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the

Matrix Valued Spherical Functions Associated to the Complex Projective Plane

Abstract The main purpose of this paper is to compute all irreducible spherical functions on G=SU(3) of arbitrary type δ∈K, where K=S(U(2)×U(1))≃U(2). This is accomplished by associating to a
...