The Lagrange interpolation polynomials in Sobolev spaces
type: Item
instance of: scholarly article
title: The Lagrange interpolation polynomials in Sobolev spaces
zbMATH Open document ID: 0905.42004
author: B. G. Gabdulkhaev
author: L. B. Ermolaeva
published in: Russian Mathematics
publication date: 1998-10-18T00:00:00Z
review text: Using the notations of the paper, being the common ones, the authors proved, among others, the following theorems. Theorem 1. For an arbitrary function \(x(s)\in W^r_2\), the interpolation polynomials \({\mathcal L}_n(x; s)\) converge in the space \(W^r_2\) for \(r> 1/2\), and for \(j= 1,2,3\) \[ E_n(x)_{(j)}\leq\| x-{\mathcal L}_nx\|_{(j)}\leq A_j E_n(x)_{(j)}, \] \[ E_n(D^rx)_2\leq\| x-{\mathcal L}_n x\|_{(j)}\leq B_j E_n(D^rx)_2, \] hold, where \(A_j\), \(B_j\) are given exactly, furthermore the norms \(\|\cdot\|_j\) are the most usable ones, being equivalent if \(r> 1/2\). Theorem 3. For any \(x(s)\in W^\ell_2[0, 2\pi]\), \(\ell\geq r> 1/2\), \({\mathcal L}_n(x; x)\) converge in \(W^r_2[0, 2\pi]\) with the rate \[ \| x-{\mathcal L}_n x\|_{W^r_2}= O(n^{r-\ell} E^T_n(x^{(\ell)})_2). \] Similar results are also proved for algebraical interpolation polynomials.
reviewed by: László Leindler
Mathematics Subject Classification ID: 42A15
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1175198
zbMATH Keywords: Lagrange interpolation polynomials
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1175198