Error of the approximation by interpolation polynomials of small degrees on n-simplices
type: Item
instance of: scholarly article
title: Error of the approximation by interpolation polynomials of small degrees on n-simplices
zbMATH Open document ID: 0734.41021
DOI: 10.1007/BF01139604
author: Yu. N. Subbotin
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: Let \(S_{h,n}\) be the set of all n-simplexes \(\Delta_ n\) in \({\mathbb{R}}^ n\) with diam \(\Delta\) \({}_ n\leq h\). If \(\Delta_ n\in S_{h,n}\), let \(C_ M^{k+1}(\Delta_ n)\) be the set of all \(C^ k\) functions on \(\Delta_ n\) with \(\| f^{(k+1)}(x)\| \leq M\) for \(x\in \Delta_ n\), and let \(p_ k(f)\) be the polynomial of degree k in n variables that interpolates \(f\in C_ M^{k+1}(\Delta_ n)\) in vertices of \(\Delta_ n\), if \(k=1\), or else in vertices of \(\Delta_ n\) and the middle points of its sides, if \(k=2\). Set \[ \alpha_{k,n}:=\sup \{\sup | f-p_ k(f)| (\Delta_ n)M^{-1}h^{-k-1}:f\in C_ M^{k+1},\Delta_ n\in S_{h,n},h>0\}. \] The author proves that 1) for each n, \(\alpha_{1,n}=n/4(n+1);\) 2) if \(n\geq 2\), then \(n\sqrt{2n}/48(n+1)\sqrt{n+1}\leq \alpha_{2,n}\leq 5n/16(n+1).\)
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 4214643
zbMATH Keywords: interpolation polynomials
zbMATH Keywords: \(C^ k\)-functions on simplexes
reviewed by: Wiesław Pleśniak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214643