Favard's interpolation problem in one or more variables
type: Item
instance of: scholarly article
title: Favard's interpolation problem in one or more variables
zbMATH Open document ID: 1015.41001
author: Thomas Kunkle
published in: Constructive Approximation
publication date: 2003-06-25T00:00:00Z
review text: The author discusses the interpolation of function values on discrete sets of data points, and the extent to which the derivatives of an interpolant can be bounded above in absolute value by a constant multiple of its necessary size, as determined by the divided differences of the data. The starting point of such investigations is a construction of \textit{J. Favard} [Sur l'interpolation. J. Math. Pures Appl., IX. Ser. 19, 281-306 (1940; Zbl 0026.01201)] that gives a complete answer to this question in the univariate case. First, the optimality of Favard's result is investigated in the article. Some improvement of the Favard's results is obtained with just a few minor changes to Favard's proof. It is shown that the \(\left(n-1\right)\)-th derivative of Favard's interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corrseponding divided differences. Then the author considers the multidimensional problems. An extension of the Favard's result to the interpolation on lattices is discussed (the author has it proved in [\textit{Th. Kunkle}, J. Approximation Theory 71, No.~1, 94-103 (1992; Zbl 0767.41007)]). The main multidimensional results concern the interpolation on tensor product grids. Some useful conditions for the interpolants are obtained. The results obtained may be very interesting for the specialists in approximation theory as well as in numerical analysis.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 65N22
zbMATH DE Number: 1885731
zbMATH Keywords: interpolation
zbMATH Keywords: Favard's interpolant
zbMATH Keywords: divided differences
zbMATH Keywords: interpolation on lattices
zbMATH Keywords: multivariate interpolation
zbMATH Keywords: interpolation on tensor product grids
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: s00365-001-0015-7
OpenAlex ID: W1977785612
cites work: Q4078977
cites work: How small can one make the derivatives of an interpolating function?
cites work: On 'best' interpolation
cites work: Sur l'interpolation
cites work: Higher-order estimates for fully nonlinear difference equations. I
cites work: Higher-order estimates for fully nonlinear difference equations. II
cites work: Q4072744
cites work: Lagrange interpolation on a lattice: Bounding derivatives by divided differences
cites work: Q4764043
DOI: 10.1007/S00365-001-0015-7
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description: scientific article; zbMATH DE number 1885731