Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

Hyperfunctional Weights for Orthogonal Polynomials

The Chebychev polynomials associated to any given moments μ_{n ∞} ⁰ are formally orthogonal with respect to the formal δ-series $$w(x)= {\sum^\infty_0}(- 1)^{n}\mu_{{n}}\delta^{(n)}(x)/n!.$$ We show that this formal weight can be a true hyperfunctional weight if its Fourier transform is a slowly increasing holomorphic function in some tubular neighborhood of the real line. It provides a unifying treatment of real and complex orthogonality of Chebychev polynomials including all classical examples and characterizes Chebychev polynomials having Bessel type orthogonality.     

Integral Norms of Trigonometric Polynomials

For trigonometric polynomials with coefficients equal to 1 or 0 in absolute value whose spectra are located on the left-hand side of binary blocks, we establish two-sided estimates of the L ₁-norm.     

Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle

This paper surveys the closely related topics included in thetitle. Emphasis is given to the parallelism between the approachusing (Perron–Carathéodory) continued fractionsto solve the trigonometric moment problem, and the alternatedevelopment that proceeds from the sequence of moments , to the linear functional µ,to the Szegö polynomials and their reciprocal and associatedpolynomials, and to the quadrature formula for µ and thesolution of the moment problem.     

Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Blumenthal''s Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B_n(x)=(x−β_n)B_n−1(x)−α_nxB_n−2(x), with positive recurrence coefficients α_n+1,β_n (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α_n→α and β_n→β and show that the zeros of B_n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.     

Ratio Asymptotics for Orthogonal Matrix Polynomials

Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind.     

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.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions

Summary. Let w _,beta (t)=(1–t)(1+t), ,>–1, denote the Jacobi weight function. For 0,<5/2 it is shown that on compact subintervals of (–1,1), the corresponding Stieltjes polynomials and their derivatives are asymptotically equal to certain Jacobi polynomials. This also leads to asymptotic representations of those weights of Gauss-Kronrod quadrature formulae, which correspond to nodes in a compact subinterval of (–1,1). On the other hand, it is demonstrated that for the parameters satisfying min(,)0 and max(,)>5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions for 02. For >3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently. Mathematics Subject Classification (1991):33C10, 33C45, 42C05, 65D32The second author is sponsored by a Heisenberg scholarship of the Deutsche Forschungsgemeinschaft.     

Estimates for Norms of Random Polynomials and Their Application

This paper contains some estimates for the integral--uniform and uniform norms of random polynomials. The integral--uniform norm introduced in [5] is a natural generalization of the maximum norm taken over a net. Some elementary properties of the integral--uniform norm are presented. Also an application of the established estimates is given.     

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

Julia Sets of Orthogonal Polynomials

Potential Analysis - For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {Pn} to...     

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,

Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.     

Computing Discrepancies of Smolyak Quadrature Rules

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as possible competition to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing theirL₂-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing theL₂-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules and Hammersley and Monte Carlo sequences.     

数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.