Hermite-Fejér interpolation of higher order with varying weights
type: Item
instance of: scholarly article
title: Hermite-Fejér interpolation of higher order with varying weights
zbMATH Open document ID: 1115.41001
DOI: 10.1007/BF02875742
author: Biancamaria Della Vecchia
author: Giuseppe Mastroianni
author: Péter Vértesi
publication date: 2007-02-02T00:00:00Z
review text: Let \(X=(x_{i,n}=x_i \in I=[-1,1]: i=1, \cdots, n; \;n \in \mathbb N)\) be an infinite matrix where each row is a set of distinct points in \(I\). For \(f \in C(I)\) the unique Hermite-Féjer interpolation polynomial \(H_{nq}(F,X,x)\) of order \(\leq nq-1\) (\(q \geq 1\) a fixed integer) is defined by \[ H^{(\ell)}_{nq}(f,X,x_{kn})=\delta_{0 \ell} f(x_{kn}), k=1, \cdots, n; \ell =1, \cdots, q-1. \] This implies that \[ H_{nq}(F,X,x) = \sum_{k=1}^n f(x_{kn}) h_{k n q}(x), \quad n = 1, 2, 3, \cdots, \] with \(h_{k n q}(x)\) polynomial of degree exactly \(nq-1\) such that \[ h^{(\ell)}_{k n q}(x_{j n})= \delta_{0 \ell} \delta_{k j}, 1 \leq k, j \leq n, 0 \leq \ell \leq q-1. \] An explicit form of \(h^{(\ell)}_{k n q}\) is \[ h^{(\ell)}_{k n q}(x) = \ell^q_{kn}(x) \sum_{i=0}^{q-1} e_{iknq}(x-x_{kn})^i, 1 \leq k \leq n, \] where \(\ell_{kn}\) are the Lagrange fundamental polynomials of degree exactly \(n-1\). When the interpolation nodes are the \(n\)th Jacobi polynomials \(P^{\alpha, \beta}(x), \alpha, \beta > -1\), then some properties of \(H_{nq}\) were investigated by \textit{R. Sakai} and \textit{P. Vértesi} [Stud. Sci. Math. Hung. 28, No. 1-2, 87--97 (1993; Zbl 0802.41006) and Stud. Sci. Math. Hung. 28, No. 3-4, 379--386 (1993; Zbl 0832.41003)]. In particular, they obtained precise estimates for the coefficients \(e_{iknq}\). In the above mentioned Jacobi case the coefficients can be estimated by the well known differential equation. However this differential equation is no longer available for the more general case of generalized Jacobi polynomials considered in the paper under review. To overcome this difficulty, the authors obtain precise estimates for the coefficients based on properties of zeros of these generalized Jacobi polynomials.
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 5121505
zbMATH Keywords: Generalized Jacobi polynomials
zbMATH Keywords: Hermite- Fejér interpolation
reviewed by: Richard A. Zalik
MaRDI profile type: Publication
cites work: Uniform asymptotic representations of orthogonal polynomials
cites work: Q4291323
cites work: Q4291351
cites work: Hermite-Fejér interpolations of higher order. I
cites work: Orthogonal polynomials based on varying Jacobi-type weights
full work available at URL: https://doi.org/10.1007/bf02875742
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5121505