Multivariate Interpolation and Conditionally Positive Definite Functions. II

Madych, W. R.; Nelson, S. A.

doi:10.2307/2008691

Cited by 103 publications

(195 citation statements)

References 2 publications

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“…Firstly, we want to establish the link between our function φ which occurs in the definition of the representer and the idea of a conditionally positive definite function. Secondly, we want to give an error estimate which is an improvement of one given in [12]. To investigate the idea of conditional positive definiteness, we utilise a definition from Gelfand and Vilenkin [5].…”

Section: Applications and Remarksmentioning

confidence: 99%

“…However, there is earlier unpublished work by Madych and Nelson which adopts very powerfully a dual viewpoint of conditionally positive definite functions linked to a variational theory. This work has been extended and enhanced by Madych and Nelson [11,12], Wu and Schaback [19] and in a series of papers by Schaback, which the reader can access through the survey [15]. The idea in all these papers is to use the theory surrounding conditionally positive definite functions to develop the correct setting for a variational approach to the above interpolation problem.…”

Section: Introductionmentioning

confidence: 99%

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Spaces of distributions, interpolation by translates of a basis function and error estimates

Light

Wayne

1999

Numerische Mathematik

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Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates. Mathematics Subject Classification (1991): 65D05

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Section: Applications and Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spaces of distributions, interpolation by translates of a basis function and error estimates

Light

Wayne

1999

Numerische Mathematik

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“…Based on earlier studies of [3], [4], [14], [15], [16], [34] among other things characterisations of the native function spaces have been surveyed in [11]. Interestingly enough, [11] proves that we do not obtain more information if we extend the definition of F ψ (Ω) to F ψ (R d ), since it can be shown that restriction of all functions from the latter to the domain Ω yields a function space that coincides with F ψ (Ω):…”

Section: Characterisations Of Native Spacesmentioning

confidence: 99%

“…-The paper [15] of Madych and Nelson provides a further example for the Gaussians ψ(x ) = e −α|x| although restricted to the univariate case d = 1 by…”

Section: Examplesmentioning

confidence: 99%

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Iske

Sonar²

1996

Numerische Mathematik

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Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Classification (1991): 65M60 41A15 41A29 41A63; 65N30 76M25 Mathematics Subject

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“…The function is conditionally positive definite of order 1 [10,21,28] on R 3 . Moreover, is in C 2 (R 3 ) [28, Theorem 4.1], so we can form its Hessian,…”

Section: Introductionmentioning

confidence: 99%

Divergence-Free RBFs on Surfaces

Narcowich

Ward

Wright

2007

J Fourier Anal Appl

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This article presents a new tool for fitting a divergence-free vector field tangent to a two-dimensional orientable surface P ∈ R 3 to samples of such a field taken at scattered sites on P. This method, which involves a kernel constructed from radial basis functions, has applications to problems in geophysics, and has the advantage of avoiding problems with poles. Numerical examples testing the method on the sphere are included.

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Multivariate Interpolation and Conditionally Positive Definite Functions. II

Cited by 103 publications

References 2 publications

Spaces of distributions, interpolation by translates of a basis function and error estimates

Spaces of distributions, interpolation by translates of a basis function and error estimates

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Divergence-Free RBFs on Surfaces

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