Construction techniques for highly accurate quasi-interpolation operators
type: Item
instance of: scholarly article
title: Construction techniques for highly accurate quasi-interpolation operators
zbMATH Open document ID: 0904.41012
author: Robert Schaback
author: Zongmin Wu
published in: Journal of Approximation Theory
publication date: 1998-02-25T00:00:00Z
review text: The authors consider univariate quasi-interpolants of the form \[ f_h(x)= \sum^{+\infty}_{-\infty} f(hj)\varphi_h(x/h- j), \] for \(x\in\mathbb{R}\) and \(h>0\), where \(\varphi_h\) is in turn a linear combination of translates \(\psi(x- jh)\) of a function \(\psi\) in \(C^\ell(\mathbb{R})\). Thus the sampling distance of the data \(f(jh)\) is actually different from the shifts used for the function \(\psi\). It is shown that it is possible to find linear combinations \(\varphi_h\) such that the order of convergence of the quasi-interpolants is only limited by the smoothness of the function \(\psi\). As the authors' technique involves discrete convolutions with \(B\)-splines, it can be generalized to the multivariate setting by using discrete convolutions with tensor-products of odd-degree \(B\)-splines.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 1121279
zbMATH Keywords: quasi-interpolation
zbMATH Keywords: order of convergence
zbMATH Keywords: Strang-Fix conditions
reviewed by: E. Quak
MaRDI profile type: Publication
OpenAlex ID: W1964130785
cites work: Q4328439
cites work: Univariate multiquadric approximation: Quasi-interpolation to scattered data
cites work: The polynomials in the linear span of integer translates of a compactly supported function
cites work: Q3469665
cites work: Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )
cites work: Fourier analysis of the approximation power of principal shift-invariant spaces
cites work: Q4731611
cites work: A Natural Formulation of Quasi-Interpolation by Multivariate Splines
cites work: Local Approximation by Certain Spaces of Exponential Polynomials, Approximation Order of Exponential Box Splines, and Related Interpolation Problems
cites work: Q4010379
cites work: Q3973499
cites work: Interpolation of scattered data: distance matrices and conditionally positive definite functions
cites work: Q3779152
cites work: Q4328451
cites work: Error estimates and condition numbers for radial basis function interpolation
cites work: Q4404383
cites work: Q3805217
cites work: Q4018018
cites work: Shape preserving properties and convergence of univariate multiquadric quasi-interpolation
cites work: Local error estimates for radial basis function interpolation of scattered data
DOI: 10.1006/JATH.1996.3101
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description: scientific article; zbMATH DE number 1121279