Showing papers on "Discrete orthogonal polynomials published in 1990"

Chebyshev polynomials : from approximation theory to algebra and number theory

Abstract: Definitions and Some Elementary Properties Extremal Properties Expansion of Functions in Series of Chebyshev Polynomials Iterative Properties and Some Remarks About the Graphs of the Tn Some Algebraic and Number Theoretic Properties of the Chebyshev Polynomials References Glossary of Symbols Index.

Orthogonal Polynomials Associated with Root Systems

Abstract: The orthogonal polynomials that are the subject of these lectures are Laurent polynomials in several variables. They depend rationally on two parameters q and t, and there is a family of them attached to each root system R. For particular values of the parameters q and t, these polynomials reduce to objects familiar in representation theory: (i) when q = t,they are independent of q and are the Weyl characters for the root system R. (ii) when q = 0 they are (up to a scalar factor) the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group G relative to a maximal compact subgroup K, such that the restricted root system of (G,K) is the dual root system R. (iii) when q and t both tend to 1, in such a way that (1 – t)/(1 – q) tends to a definite limit k , then (for certaion values of k) our polynomials guive the values of zonal spherical functions on a real (compact or noncompact) symmetric space G/K arising from finite-dimensional spherical representations of G, that is to say representations having a non zero K-fixed vector. Here the root system R is the restricted root system of G/K, and the parameter k is half the root multiplicity (assumed to be the same for all restricted roots).

Characterization Theorems for Orthogonal Polynomials

Abstract: We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.

Orthogonal polynomials : theory and practice

Abstract: Characterization Theorems for Orthogonal Polynomials.- Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics.- Orthogonal Polynomials, Pade Approximations and Julia Sets.- The Three Term Recurrence Relation and Spectral Properties of Orthogonal Polynomials.- On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal.- A Survey on the Theory of Orthogonal Systems and Some Open Problems.- Orthogonal Polynomials and Functional Analysis.- Using Symbolic Computer Algebraic Systems to Derive Formulas Involving Orthogonal Polynomials and Other Special Functions.- Computational Aspects of Orthogonal Polynomials.- The Recursion Method and the Schroedinger Equation.- Birth and Death Processes and Orthogonal Polynomials.- Orthogonal Polynomials in Connection with Quantum Groups.- The Approximate Approach to Orthogonal Polynomials for Weights on (-?,?).- Orthogonal Polynomials Associated with Root Systems.- Some Extensions of the Beta Integral and the Hypergeometric Function.- Orthogonal Matrix Polynomials.- Orthogonal Polynomial from a Complex Perspective.- Nth Root Asymptotic Behavior of Orthonormal Polynomials.- An Introduction to Group Representations and Orthogonal Polynomials.- Asymptotics for Orthogonal Polynomials and Three - Term Recurrences.- List of of Participants.- Scientific Program.

Sur la suite de polynômes orthogonaux associée à la forme u=δ c +λ(x−c) −1 L

Abstract: We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

Wavefront fitting with discrete orthogonal polynomials in a unit radius circle

Abstract: Zernike polynomials have been used for some time to fit wavefront deformation measurements to a two-dimensional polynomial. Their orthogonality properties make them ideal for this kind of application. The typical procedure consists of first obtaining the fitting using x-y polynomials and then transforming them to Zernike polynomials by means of a matrix multiplication. Here, we present a new method for making this fitting faster by using a set of orthogonal polynomials on a discrete base of data points on a unitary circle.

Quantum 2-spheres and big q-Jacobi polynomials

Abstract: Orthogonal bases for the algebras of functions of Podles' quantum 2-spheres are explicitly determined in terms of bigq-Jacobi polynomials. This gives a group-theoretic interpretation of the symmetric bigq-Jacobi polynomials and the symmetricq-Hahn polynomials.

On the Construction of Irreducible Self-Reciprocal Polynomials Over Finite Fields

Abstract: The transformationf(x)↦f Q x≔deg(f) f(x + 1/x) for f(x)∈ $$\mathbb{F}_q [x]$$ is studied. Simple criteria are given for the case that the irreducibility off is inherited by the self-reciprocal polynomialf Q . Infinite sequences of irreducible self-reciprocal polynomials are constructed by iteration of thisQ-transformation.

Asymptotics for Orthogonal Polynomials and Three-Term Recurrences

Abstract: 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.

Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems

Abstract: Local approximation order to smooth complex valued functions by a finite dimensional space #7B-H, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical #7B-H is based here on the identification of its dual with a certain space #7B-P of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from #7B-P

Fast decreasing polynomials

Abstract: Matching two-sided estimates are given for the minimal degree of polynomialsP satisfyingP(0)=1 and ¦P(x)|≤exp(−ϕ (¦x¦)),x ∈ [−1,1], whereϕ is an arbitrary, in [0, 1], increasing function. Besides these fast decreasing polynomials we also consider bell-shaped polynomials and polynomials approximating well the signum function.

Orthogonal Polynomials from a Complex Perspective

Abstract: Complex function theory and its close companion — potential theory — provide a wealth of tools for analyzing orthogonal polynomials and orthogonal expansions. This paper is designed to show how the complex perspective leads to insights on the behavior of orthogonal polynomials. In particular, we discuss the location of zeros and the growth of orthogonal polynomials in the complex plane. For some of the basic results we provide proofs that are not typically found in the standard literature on orthogonal polynomials.

Computational Aspects of Orthogonal Polynomials

Abstract: Our concern here is with computational methods for generating orthogonal polynomials and related quantities. We focus on the case where the underlying measure of integration is nonclassical. The main problem, then, is that of computing the coefficients in the basic recurrence relation satisfied by orthogonal polynomials. Two principal methods are considered, one based on modified moments, the other on inner product representations of the coefficients. The first method is the more economical one, but may be subject to ill-conditioning. A study is made of the underlying reasons for instability. The second method, suitably implemented, is more widely applicable, but less economical. A number of problem areas in the physical sciences and in applied mathematics are described where these methods find useful applications.

Computer generation of distance polynomials of graphs

Abstract: A computer program is developed to compute distance polynomials of graphs containing up to 200 vertices. The code also computes the eigenvalues and the eigenvectors of the distance matrix. It requires as input only the neighborhood information from which the program constructs the distance matrix. The eigenvalues and eigenvectors are computed using the Givens-Householder method while the characteristic polynomials of the distance matrix are constructed using the codes developed by the author before. The newly developed codes are tested out on many graphs containing large numbers of vertices. It is shown that some cyclic isospectral graphs are differentiated by their distance polynomials although distance polynomials themselves are in general not unique structural invariants.

Inhomogeneous basis set of symmetric polynomials defined by tableaux.

Abstract: A basis set of inhomogeneous symmetric polynomials, denoted by tlambda(z), where z = (z1,....,zn) and lambda = [lambda1,...., lambdan] is a partition, is defined in terms of Young-Weyl standard tableaux or, equivalently, in terms of Gel'fand-Weyl patterns. A number of significant properties of these polynomials are given (together with outlines of proofs) and compared with properties of the well-known basis set of Schur functions, which are homogeneous polynomials. The basis of the ring of symmetric polynomials defined here is shown to be natural for the expansion of inhomogeneous symmetric functions constructed from rising factorials.

Orthogonal Matrix Polynomials

Abstract: An overview is given of some classical and recent results concerning zeros of orthogonal matrix polynomials on the unit circle. The basic questions are: How these zeros are located in the complex plane? Conversely, what conditions on the location of the zeros of a given matrix polynomial ensure that the polynomial is orthogonal?

On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal Processing Applications

Abstract: The aim of this contribution is first to show how the Szego theory of orthogonal polynomials on the unit circle is intimately related to the celebrated Levinson algorithm, which is commonly used in digital signal processing (DSP) applications to solve various least-squares problems. A computationally more efficient substitute for the Levinson algorithm, termed the split Levinson algorithm, has recently been proposed in the DSP literature. In the case of real data, this new algorithm can be interpreted naturally in the framework of a well-defined one-to-one correspondence between the families of real Szego polynomials and the families of polynomials orthogonal on the interval [-1, 1] with respect to a symmetric measure. More generally, the philosophy underlying the split Levinson algorithm opens the door to an interesting “tridiagonal approach” to the theory of complex Szego polynomials, nonnegative definite Hermitian Toeplitz matrices, and related algebraic and function theoretic questions. Some of the main topics of this new mathematical framework are briefly reviewed and are shown on specific examples to be of particular interest in DSP applications.

On the coefficients of integrated expansions of ultraspherical polynomials

Abstract: The problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered. A formula is proved for the ultraspherical polynomials, of which the Chebyshev and Legendre polynomials are important special cases.

Orthogonal Polynomials and Functional Analysis

Abstract: This paper studies the measure of orthogonality for a system of polynomials defined by a three term recursion formula, using the techniques of operator theory and functional analysis. Spectral properties of self-adjoint operators and compact operators, perturbation theorems, and commutator equations are used in the development of the ideas.

An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field

Abstract: We establish a formula for the number of irreducible polynomialsf(x) over the binary fieldF2 of given degreen ≧ 2 for which the coefficient ofx n-1 and ofx is equal to 1. This formula shows that the number of such polynomials is positive for alln ≧ 2 withn ≠ 3. These polynomials can be applied in a construction of irreducible self-reciprocal polynomials overF2 of arbitrarily large degrees.