Hermite interpolation at the zeros of certain Freud-type orthogonal polynomials
type: Item
instance of: scholarly article
title: Hermite interpolation at the zeros of certain Freud-type orthogonal polynomials
zbMATH Open document ID: 0782.41005
DOI: 10.1007/BF00051640
published in: Acta Mathematica Hungarica
publication date: 1993-04-01T00:00:00Z
review text: Assume that \[ A:=\{x_{kn},\;k=1,\ldots,n,\;n=1,2\ldots\}, \tag{1.1} \] \[ B:=\{y_{kn},\;k=1,\ldots,n,\;n=1,2\ldots\}, \tag{1.2} \] be triangular schemes of real numbers and \(f: \mathbb{R}\to\mathbb{R}\). The Hermite interpolation polynomial \(H_ n(f,x):=H_ n(A,B,f,x)\) is the unique polynomial of degree at most \(2n-1\) which satisfies (1.3) \(H_ n(f,x_{kn})=f(x_{kn})\), \(H_ n'(f,x_{kn})=y_{kn}\), \(k=1,\ldots,n\), \(n=1,2\ldots\). When all the \(y_{kn}\)'s are equal to zero, \(H_ n(A,B,f)\) is called the Hermite-Fejér interpolation polynomial. In 1930, L. Fejér proved the following theorem [cf. Constructive function Theory by \textit{I. P. Natanson} (1965; Zbl 0178.397)]. Theorem 1.1 Let \(f\) be continuous on \([-1,1]\), \(x_{kn}=\cos\bigl({2k-1\over 2n}\pi\bigr)\), \(k=1,\ldots,n\), \(n=1,2,\ldots\) and \[ \lim_{n\to\infty}\max_{1\leq k\leq n}\left\{{\log n\over n}\sqrt{1-x^ 2_{kn}}| y_{kn}|\right\}=0. \] Then the sequence \(\bigl\{H_ n(A,B,f)\bigr\}\) converges to \(f\) uniformly on \([- 1,1]\) as \(n\to\infty\). \textit{R. Bojanic} [Proc. Conf. construct Theory Functions (Approximation Theory) 1969, 69-76 (1972; Zbl 0259.41001)] proved the following theorem. Theorem 1.2 Let \(f: [-1,1]\to\mathbb{R}\) be continuous on \([-1,1]\) and for \(\delta>0\), we put \[ \omega(f,\delta):=\max\bigl\{| f(x)-f(t)|,\;| x- t|\leq\delta,\;x,t\in[-1,1]\bigr\}. \tag{1.4} \] Let \(\{x_{kn}\}\) be as in Theorem 1.1 and \(y_{kn}\)'s be all equal to zero. Then for \(x\in[- 1,1]\), \[ | H_ n(f,x)-f(x)|\leq{c\over n}\sum^ n_{k=1}\omega\left(f,{1\over k}\right),\quad n=1,2,\ldots \tag{1.5} \] where \(c\) is an absolute constant. ``With the aid of recent research in the theory of orthogonal polynomials, it has now become possible to obtain the analogues of Theorems 1.1 and 1.2, when the \(x_{kn}\)'s are the zeros of polynomials orthogonal on the whole real line with respect to a weight function of the form \(w^ 2_ Q(x):=\exp\bigl(-2Q(x)\bigr)\) satisfying various technical conditions.…
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 149633
zbMATH Keywords: Hermite interpolation polynomial
zbMATH Keywords: Hermite-Fejér interpolation polynomial
zbMATH Keywords: rate of convergence
zbMATH Keywords: modulus of continuity
zbMATH Keywords: weighted polynomial approximation
author: H. N. Mhaskar
author: Yuan Xu
reviewed by: Y. Sitaraman
MaRDI profile type: Publication
cites work: Notes on interpolation. VII
cites work: Estimates of asymmetric Freud polynomials on the real line
cites work: Estimates of the Hermite and the Freud polynomials
cites work: Q3817079
cites work: On Markov-Bernstein-type inequalities and their applications
cites work: Q4120972
cites work: Q4097662
cites work: Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line
cites work: Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules
cites work: Weights on the real line that admit good relative polynomial approximation, with applications
cites work: Strong asymptotics for extremal polynomials associated with weights on \({\mathbb{R}}\)
cites work: Extensions of the Dirichlet-Jordan criterion to a general class of orthogonal polynomial expansions
cites work: Weighted polynomial approximation
cites work: Bounds for certain Freud-type orthogonal polynomials
cites work: The rate of convergence of expansions in Freud polynomials
cites work: Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
cites work: Where Does the L p -Norm of a Weighted Polynomial Live?
cites work: Q5343522
cites work: Géza Freud, orthogonal polynomials and Christoffel functions. A case study
cites work: Mean convergence of Lagrange interpolation. I
cites work: ON ASYMPTOTIC PROPERTIES OF POLYNOMIALS ORTHOGONAL ON THE REAL AXIS
cites work: On the convergence of Hermite-Fejér interpolation for the Laguerre abscissas
full work available at URL: https://doi.org/10.1007/bf00051640
OpenAlex ID: W2055578679
description: scientific article; zbMATH DE number 149633