Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481).     

Modified Moments and Matrix Orthogonal Polynomials

In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified Chebyshev algorithm. Using the concept of matrix biorthogonality, we extend this algorithm to the vector case.     

Ratio Asymptotics for Polynomials Orthogonal on Arcs of the Unit Circle

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of lim _n |Φ _n (0)| and {lim _n [ Φ _n+1 (0)/ Φ _n (0)] , where denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered. December 27, 1996. Date revised: October 14, 1997. Date accepted: December 22, 1997.     

Laurent 多项式及其零点

本文给出了测度dψ为强分布的一个必要条件,并得到了dψ为强分布时的Laurent多项式最大零点的一个表示。     

Stieltjes Polynomials and Related Quadrature Formulae for a Class of Weight Functions

Consider a (nonnegative) measure with support in the interval such that the respective orthogonal polynomials, above a specific index , satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index , have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval (under an additional assumption on ), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.

     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Riesz's Theorem for Orthogonal Matrix Polynomials

We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of V for which the matrix polynomials are dense in the corresponding ² space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set. May 20, 1997. Date revised: January 8, 1998.     

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.     

A Semi-Conjugate Matrix Boundary Value Problem for General Orthogonal Polynomials on an Arbitrary Smooth Jordan Curve

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points { _k } _{k = 1} included in a compact subset of the unit disk, we consider the orthogonal rational functions _n that are obtained by orthogonalization of the sequence { 1, z / ₁, z ² / ₂, ... } where , with respect to the inner product In this paper we discuss the behaviour of _n (t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all _k = 0.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

Difference Equations and Discriminants for Discrete Orthogonal Polynomials

We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order difference equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application we give explicit evaluations of the discrete discriminants of the Meixner and the Hahn polynomials. A difference analogue of the Bethe Ansatz equations is also mentioned.Research partially supported by NSF grant DMS 99-70865     

Orthogonal polynomials for refinable linear functionals

A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.

     

Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid

We construct an expansion of a discrete function in the form of a mixed series of Chebyshev polynomials. We obtain estimates of the approximation error of the function and its derivatives.     

Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its, and Kitaev.