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Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
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Cited by 34 (5 self)
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. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
A Survey on Spherical Spline Approximation
 Surveys Math. Indust
, 1997
"... Spline functions that approximate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that ..."
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Cited by 14 (1 self)
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Spline functions that approximate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A convergence theorem containing explicit constants yields a usable error bound. Our survey ends with the discussion of spherical splines in geodetically relevant pseudodifferential equations. (submitted to "Surveys on Mathematics for Industry") AMS classification: 41A05, 43A90, 65D07, 86A30 Keywords: spherical splines, scattered data interpolation, smoothing, geoid determination Contents 1 Introduction 3 2 Preliminaries 4 3 Sobolev Spaces and Pseudodifferential Operators 7 4 Sobolev Lemma and Reproducing Kernel Sobolev Spaces 10 5 Examples of Radial Basis Functions 13 5.1 Green's Kernels Corresponding to Iterated Beltrami Oper...
Hermite Interpolation with Radial Basis Functions on Spheres
 Adv. Comput. Math
, 1999
"... . We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly po ..."
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Cited by 7 (1 self)
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. We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented. AMS classification: 41A05, 41A63, 42A82. Key words and phrases: Spherical interpolation, Hermite interpolation, Radial basis functions. 1. Introduction In 1975 R. Hardy mentioned the possibility of using multiquadric basis functions for Hermite interpolation (see [10], or the survey paper [11]). This problem, however, was not further investigated until the paper [29] by Wu appeared. Since then, the interest in this topic seems to have ...
Wavelet Approximations on Closed Surfaces and Their Application to Boundaryvalue Problems of Potential Theory
 Problems of Potential Theory, Mathematical Methods in the Applied Sciences
"... Wavelets on closed surfaces in Euclidean space R 3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations o ..."
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Cited by 3 (2 self)
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Wavelets on closed surfaces in Euclidean space R 3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of function values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of bandlimited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem. Abbreviated title: Wavelets and Boundaryvalue Problems AMS Subject classificat...
Constructive Approximation and Numerical Methods in Geodetic Research Today  an Attempt at a Categorization Based on an Uncertainty Principle
 JOURNAL OF GEODESY
, 1999
"... This review article reports current activities and recent progress on constructive approximation and numerical analysis in physical geodesy. The paper focuses on two major topics of interest, namely trial systems for purposes of global and local approximation and methods for adequate geodetic applic ..."
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Cited by 3 (1 self)
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This review article reports current activities and recent progress on constructive approximation and numerical analysis in physical geodesy. The paper focuses on two major topics of interest, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantication of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.