Generalized Christoffel functions for power orthogonal polynomials

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.     

Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.     

The Christoffel Function for Orthogonal Polynomials on a Circular Arc

For the special type of weight functions on circular arc we study the asymptotic behavior of the Christoffel kernel off the arc and of the Christoffel function inside the arc. We prove Totik's conjecture for the Christoffel function corresponding to such weight functions.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Szegö quadrature formulas for certain Jacobi-type weight functions

In this paper we are concerned with the estimation of integrals on the unit circle of the form by means of the so-called Szegö quadrature formulas, i.e., formulas of the type with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions related to the Jacobi functions for the interval nodes and weights in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.     

On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands

In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up to degree as high as possible are constructed in order to approximate 2π-periodic weighted integrals. For this purpose, certain bi-orthogonal systems of trigonometric functions are introduced and their most relevant properties studied. Some illustrative numerical examples are also given. The paper completes the results previously given by Szeg? in Magy Tud Akad Mat Kut Intez Közl 8:255–273, 1963 and by some of the authors in Annales Mathematicae et Informaticae 32:5–44, 2005.     

Asymptotic Gauss–Jacobi quadrature error estimation for Schwarz–Christoffel integrals

Numerical conformal mapping packages based on the Schwarz–Christoffel formula have been in existence for a number of years. Various authors, for good reasons of practical efficiency, have chosen to use composite n-point Gauss–Jacobi rules for the estimation of the Schwarz–Christoffel path integrals. These implementations rely on an ad hoc, but experimentally well-founded, heuristic for selecting the spacing of the integration end-points relative to the position of the nearby integrand singularities. In the present paper we derive an explicitly computable estimate, asymptotic as n→∞, for the relevant Gauss–Jacobi quadrature error. A numerical example illustrates the potential accuracy of the estimate even at low values of n. It is apparent that the error estimate will allow the adaptive construction of composite rules in a manner that is more efficient than has been possible hitherto.     

Asymptotic behavior of Müntz-Christoffel functions at the endpoints

We establish asymptotics for Christoffel functions of Müntz systems at the endpoints x=0 and x=1 of [0,1], assuming that there exists a ρ>0, such that the Müntz exponents {λ_k} satisfy

     

Askey-Wilson type functions with bound states

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grünbaum. 2000 Mathematics Subject Classification Primary—33D45, 37K10, 14H70, 39A70, 39A13     

Orthogonal polynomials and Gaussian quadrature for refinable weight functions

We show how to compute the modified moments of a refinable weight function directly from its mask in O(N²n) rational operations, where N is the desired number of moments and n the length of the mask. Three immediate applications of such moments are:

• the expansion of a refinable weight function as a Legendre series;

• the generation of the polynomials orthogonal with respect to a refinable weight function;

• the calculation of Gaussian quadrature formulas for refinable weight functions.

In the first two cases, all operations are rational and can in principle be performed exactly.

Gaussian Quadrature Rules with Simple Node-Weight Relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.     

Generating functions and companion symmetric linear functionals

In this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that . In some cases we can deduce explicitly the expression for the generating function

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m ⁴ n ^–6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday     

Some results about numerical quadrature on the unit circle

In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.Finally, numerical experiments are carried out.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.     

Special matrix functions: characteristics,achievements and future directions

ABSTRACT

In recent years, special matrix functions and polynomials of a real or complex variable have been in a focus of increasing attention leading to new and interesting problems. In this work, we present matrix space analogues to generalized some functions and polynomials in the framework of matrix setting. Many of the special matrix functions and polynomials are constructed along standard procedures. Recently published papers are also surveyed and we list the most essential ones.     

Some error expansions for Gaussian quadrature

Complex-variable methods are used to obtain some expansions in the error in Gaussian quadrature formulae over the interval [– 1, 1]. Much of the work is based on an approach due to Stenger, and both circular and elliptical contours are used. Stenger's theorem on monotonicity of convergence of Gaussian quadrature formulae is generalized, and a number of error bounds are obtained.     

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.     

Some results involving Hermite-base polynomials and functions using operational methods

This paper is an attempt to stress the usefulness of the operational methods in the theory of special functions. Using operational methods, we derive summation formulae and generating relations involving various forms of Hermite-base polynomials and functions.