Generalised sk-Spline Interpolation on Compact Abelian Groups

Levesley, Jeremy; Kushpel, Alexander

doi:10.1006/jath.1997.3267

Cited by 12 publications

(12 citation statements)

References 18 publications

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“…In the case when G=(SO(2)) d , the zonal kernels on the torus are convolution kernels, and it is clear the T k is non-singular on H n if and only if a : {0 for each |:| =n. This agrees with the result of Levesley and Kushpel [3].…”

Section: Examples Of Zonal Kernel Operatorssupporting

confidence: 79%

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The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

Ragozin

Levesley

2000

Journal of Approximation Theory

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Compact manifolds embedded in Euclidean space which have a transitive group G of linear isometries, such as the spheres with the rotation group or the``flat'' tori with the group of rotations in each coordinate direction, admit a natural notion of a continuous G-invariant kernel function k(x, y), which generalizes the idea of a radial or distance-dependent function on the spheres and tori. In connection with a study of quasi-interpolation on these spaces, we have reproved and extended results of Sun for the spheres to characterize those kernels for which the span of the translates, a n k(x, y n ), is dense in the continuous functions. The essence of the characterization is that the integral operator with G-invariant kernel k(x, y) must be non-singular when restricted to the space of n th degree polynomial functions. This requires that the polynomials be invariant under all such linear operators, which is true for many compact homogeneous M including the spheres, tori, and others. In fact the non-singularity must hold only on any finite-dimensional space of zonal polynomials, those which are pointwise fixed by the subgroup of all isometries fixing a single point on M. In practical terms this later condition is verified by choosing one point on the manifold (the north pole on the spheres or the identity element on the flat tori), picking some basis for the polynomials of

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Section: Examples Of Zonal Kernel Operatorssupporting

confidence: 79%

“…For spheres or tori characterization of K has been achieved previously by the use of facts about spherical harmonics or multiple Fourier series [3,7,8]. Our aim here is to simplify and unify these cases by dispensing with these facts and using only extremely elementary facts from geometry, analysis, and approximation theory.…”

Section: Introductionmentioning

confidence: 98%

The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

Ragozin

Levesley

2000

Journal of Approximation Theory

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“…Kernels used in such an interpolation procedure go by various names in the literature: periodic radial basis functions, circular basis functions, and periodic sksplines, to name a few [14,12,13]. To guarantee the existence and uniqueness of such an interpolant, we will use conditionally positive definite kernels (defined below).…”

Section: Interpolation With Periodic Basis Functionsmentioning

confidence: 99%

Order-preserving derivative approximation with periodic radial basis functions

Fuselier

Wright

2014

Adv Comput Math

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In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the m th derivative is obtained by m successive applications of the operator "interpolate, then differentiate" -this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a sufficient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates all derivatives with the same rate of convergence. We show that thin-plate spline, power function, and Matérn kernels restricted to the circle all satisfy this condition, and numerical evidence is provided to show that this phenomena occurs for some other popular RBF kernels. Finally, we consider possible extensions to higher-dimensional periodic domains by numerically studying the convergence of an iterated method for approximating the surface Laplace (Laplace-Beltrami) operator using RBF interpolation on the unit sphere and a torus.

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“…A fundamental problem of approximation theory on S 2 which has attracted a lot of attention and remained opened for a long time is to construct in an explicit form the best possible method of recovery of functions f from classical Sobolev's classes W γ ∞ (S 2 ) in L ∞ (S 2 ) using their values on an "optimal" set of points X N := {x k } N k=1 ⊂ S 2 [23]. Remark that using grided data points on the torus, T d , it is not possible to obtain the best possible order of convergence on Sobolev's classes and optimal methods of reconstruction (interpolation) of functions from these sets are connected with the theory of uniform distribution of sequences [31,9,24]. The fundamental number theoretic concepts in this area were advanced in [15,16].…”

Section: Introductionmentioning

confidence: 99%