Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.Comment: 20 pages, 5 figures. Keywords: Charlier polynomials, Laguerre polynomials, Jacobi polynomials, asymptotic expansions, saddle point methods, two-points Taylor expansion

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. As an application, we see that the resulting bases can be used to interpolate functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+ \epsilon^2}$, or $1/x$, with exponential convergence

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson--Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets

Generalized translation operator and approximation in several variables

Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted $L^p$ spaces. In one variable, this becomes the generalized translation operator for the Gegenbauer polynomial expansions.Comment: 22 pages, 7th International Symposium on Orthogonal Polynomials and Special Functions, Copenhagen, August 200

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the Riemann-Hilbert problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order $1/n$ terms in the expansions. A critical step in the analysis of the Riemann-Hilbert problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order.Comment: 31 pages, 6 figures, 21 reference

A formula for polynomials with Hermitian matrix argument

AbstractWe construct and study orthogonal bases of generalized polynomials on the space of Hermitian matrices. They are obtained by the Gram–Schmidt orthogonalization process from the Schur polynomials. A Berezin–Karpelevich type formula is given for these multivariate polynomials. The normalization of the orthogonal polynomials of Hermitian matrix argument and expansions in such polynomials are investigated

Some optimality conditions for Chebyshev expansions

AbstractIn this paper we investigate conditions under which approximation to continuous functions on [−1, 1] by series of Chebyshev polynomials is superior to approximation by other ultraspherical orthogonal expansions. In particular we derive conditions on the Chebyshev coefficients which guarantee that the Chebyshev expansion of the corresponding functions converges more rapidly than expansions in Legendre polynomials or Chebyshev polynomials of the second kind