Three conjectures on Shepard interpolatory operators
type: Item
instance of: scholarly article
title: Three conjectures on Shepard interpolatory operators
zbMATH Open document ID: 0905.41005
author: Tingfan Xie
author: Renjiang Zhang
author: S. P. Zhou
published in: Journal of Approximation Theory
publication date: 1998-10-28T00:00:00Z
review text: For \(f\in C[0,1] \), the Shepard interpolatory operator \(S_{n,\lambda} \) has the images \[ (S_{n,\lambda}f)(x)=\frac{ \sum_{k=0}^{n}| x-k/n| ^{- \lambda}f(k/n) } {\sum_{k=0}^{n} | x-k/n| ^{- \lambda}}, \] where \(n=1,2,\dots; \lambda \geq 1\). The main results are the following: Theorem 1: There is at least a function \( f , f\in C^1[0,1] \) such that \[ \| f-S_{n,2}f\| =O(1/n), \] while \(\int_{0}^{1}{ t^{-1}\omega(f',t)dt} = \infty . \) Theorem 2: For \(1< \lambda < 2 \), \[ \| f-S_{n,\lambda }f\| = o (n^{1 - \lambda}) \text{ implies } f=\text{const.} \] Moreover, there is a function \(f\in C[0,1] \) such that \[ \| f-S_{n,\lambda}f\| =O(n^{1- \lambda}), \] while \(\int_{0}^{1}t^{-1}\omega(f,t)dt = \infty\). Theorem 3: If \(\| f-S_{n,1} f\|= o (\log^{-1}{n})\), then \(f=\text{const.}\) Also, there is a function \(f\in C[0,1] \) such that \[ \| f-S_{n,1}f\| =O( \log^{-1}{n}), \] while \( \int_{0}^{1}t^{-1}\omega (f,t)dt = \infty\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A40
zbMATH DE Number: 1197061
zbMATH Keywords: interpolation
zbMATH Keywords: approximation by positive operators
zbMATH Keywords: saturation
zbMATH Keywords: Shepard interpolatory operator
reviewed by: Alexandru Lupaş
MaRDI profile type: Publication
full work available at URL: jath.1998.3180
OpenAlex ID: W1991668590
cites work: Q5758631
cites work: Estimates of the Shepard interpolatory procedure
cites work: On an interpolatory operator and its saturation
cites work: On a saturation problem
cites work: Q4017993
cites work: A constructive method related to generalizations of the uniform boundedness principle and its applications
Wikidata QID: Q122971722
DOI: 10.1006/JATH.1998.3180
Recommended article: Q4017993
Recommended article: Q4732844
Recommended article: Q3071192
Recommended article: Q4877685
Recommended article: Q3416434
Recommended article: Q4384587
Recommended article: On the approximation order of triangular Shepard interpolation
Recommended article: Q5431285
Recommended article: ON OPERATOR INTERPOLATION PROBLEMS
Recommended article: On the norm of certain interpolating operators
description: scientific article; zbMATH DE number 1197061