Walter Van Assche

Asymptotics for orthogonal polynomials

Orthogonal polynomials on a compact set.- Asymptotically periodic recurrence coefficients.- Probabilistic proofs of asymptotic formulas.- Orthogonal polynomials on unbounded sets.- Zero distribution and consequences.- Some applications.

Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 x×2 matrix functions) associated with a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on orthogonal polynomials, in particular strong asymptotics which hold uniformly in the complex plane. In this paper we will show that a similar Riemann-Hilbert problem (for (r + 1) × (r + 1) matrix functions) is associated with multiple orthogonal polynomials. We show how this helps in understanding the relation between two types of multiple orthogonal polynomials and the higher order recurrence relations for these polynomials. Finally we indicate how an extremal problem for vector potentials is important for the normalization of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials.

Some classical multiple orthogonal polynomials

Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system of several functions. We describe seven families of multiple orthogonal polynomials which have he same flavor as the very classical orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some open research problems and some applications.

Orthogonal matrix polynomials and applications

Orthogonal matrix polynomials, on the real line or on the unit circle, have properties which are natural generalizations of properties of scalar orthogonal polynomials, appropriately modified for the matrix calculus. We show that orthogonal matrix polynomials, both on the real line and on the unit circle, appear at various places and we describe some of them. The spectral theory of doubly infinite Jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line. Scalar orthogonal polynomials with a Sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line. Orthogonal matrix polynomials on the unit circle are related to unitary block Hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing. Finally we show how orthogonal matrix polynomials can be used for Gaussian quadrature of matrix-valued functions. Parallel algorithms for this purpose have been implemented (using PVM) and some examples are worked out.

Asymptotics for Orthogonal Polynomials and Three-Term Recurrences

It is often desirable to obtain (asymptotic) properties of orthogonal polynomials and the measure with respect to which these polynomials are orthogonal. All orthogonal polynomials on the real line (with a positive Borel measure) satisfy a three term recurrence relation. We give a survey showing how properties of the recurrence coefficients reveal properties of the corresponding orthogonal polynomials.

Orthogonal polynomials, associated polynomials and functions of the second kind

Abstract A survey is given of the interaction between orthogonal polynomials, associated polynomials and functions of the second kind with an emphasis on asymptotic results. Various formulas are presented in a unified way in terms of Wronskians of solutions of linear recurrence relations. Some of these formulas are classical and go back to the previous century, but usually they are hard to locate in the literature. Some new formulas are also given, in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials.

Polynomial interpolation and Gaussian quadrature for matrix-valued functions

Abstract The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision.

Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.

Nearest neighbor recurrence relations for multiple orthogonal polynomials

Abstract We show that multiple orthogonal polynomials for r measures ( μ 1 , … , μ r ) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices n → ± e → j , where e → j are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of the measures μ j . We show how the Christoffel–Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.

Asymptotic properties of orthogonal polynomials from their recurrence formula, II

Abstract In this paper it is supposed that the coefficients in the recurrence formula for orthogonal polynomials have finite limits as the index goes to infinity over the set of even and odd integers. The asymptotic behavior of the ratio of two contiguous polynomials and the limiting zero distribution are discussed. Applications to quadrature formulas are given.