A generalization of Jackson's inequality
type: Item
instance of: scholarly article
title: A generalization of Jackson's inequality
zbMATH Open document ID: 0537.41008
DOI: 10.1016/0021-9045(81)90039-3
published in: Journal of Approximation Theory
publication date: 1981-01-01T00:00:00Z
review text: In this article the following assertions are proved: Theorem 1. Let \(S_{k,n}\) be the space of smooth splines of degree k-1 (i.e., \(S_{k,n}\subset C^{k-2})\) with n equally spaced knots on the interval [0,1]. Then \((1)\quad \inf_{s\in S_{k,n}}\| f-s\|_ p\leq |(C\cdot 1/(n+k))\| f'\|_ p,\) where C is an absolute constant less than 10. Theorem 2. Let \(S_{k,t}\) be the space of smooth splines on a mesh \(t=\{0=t_ 0<t_ 1<...<t_ n<t_{n+1}=1\}\). Then \(\inf_{s\in S_{k,t}}\| f-s\|_ p\leq C'(({\bar \Delta})^{- 1}+k)^{-j}\| f^{(j)}\|_ p,\) \(k\geq j\), where C' only depends on j, and \({\bar \Delta}=\max_{0\leq i\leq n}(t_{i+1}-t_ i).\) In these theorems \(f^{(j)}\) denotes the derivative of order j of the function f, 1\(\leq p\leq \infty\), and \(\| \cdot \|\) denotes the usual norm of the space \(L_ p[0,1]\). For \(n=0\), (1) reduces to Jackson's well-known inequality for polynomial approximation.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3853666
zbMATH Keywords: Jackson's inequality
zbMATH Keywords: smooth splines
author: Klaus Höllig
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: 0021-9045(81)90039-3
OpenAlex ID: W2050723673
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description: scientific article; zbMATH DE number 3853666