Convergence rate for trigonometric interpolation of non-smooth functions
type: Item
instance of: scholarly article
title: Convergence rate for trigonometric interpolation of non-smooth functions
zbMATH Open document ID: 0806.42002
author: Yuan Xu
author: Jürgen Prestin
published in: Journal of Approximation Theory
publication date: 1995-02-12T00:00:00Z
review text: For a \(2\pi\)-periodic bounded function \(f\) which is measurable on \([0,2\pi]\), the authors obtain estimates for the order of approximation of \(f\) in \(L_ p\) by trigonometric interpolation polynomials. Let \(n,q\in \mathbb{N}\) and \(0< m_ 1<\cdots< m_ q\) be distinct integers. For given \(n\), \(q\), and \(f\) we consider the interpolation polynomial \(F_ n f\) of order at most \(M= M(n,q)\), which is defined by \[ F_ n(f;\theta_{kn})= f(\theta_{kn}),\;k=0,\dots,n- 1,\quad F^{(j)}_ n(f;\theta_{kn})= 0,\;j= m_ 1,\dots,m_ q,\;k= 0,\dots,n- 1, \] where \(\theta_{kn}= 2k\pi/n\), \(k=0,\dots,n- 1\). One of the main results of the paper is the following inequality \[ \| F_ n f- f\|_{L_ p}\leq C_ p(\widetilde E_ n(f)_ p+ \omega_{m_ 1}(f; 1/n)_ p), \] where \(\widetilde E_ n(f)_ p\), \(1< p<\infty\), is the best one- sided approximation of \(f\) in \(L_ p\) by trigonometric polynomials of order at most \(n\) and \(\omega_{m_ 1}(f;\delta)_ p\) is the \(m_ 1\)th order \(L_ p\) modulus.
Mathematics Subject Classification ID: 42A15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 597640
zbMATH Keywords: trigonometric interpolation
zbMATH Keywords: trigonometric approximation
zbMATH Keywords: degree of approximation
zbMATH Keywords: trigonometric interpolation polynomials
zbMATH Keywords: one-sided approximation
MaRDI profile type: Publication
full work available at URL: jath.1994.1037
OpenAlex ID: W2014374293
DOI: 10.1006/JATH.1994.1037
description: scientific article; zbMATH DE number 597640