We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

. 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 multi-resolution methods. In addition, we briefly discuss sphere-like 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...

After a series of application papers have proven the approach to be numerically effective, this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with (possibly radial) basis functions. 0 Introduction We consider a general class of boundary or initial value problems for partial differential equations: Lu = f in\Omega ae IR d L : W\Omega ! L\Omega Bu = g in @\Omega B : W\Omega !W @\Omega : (0.1) Here, the operator L is a linear partial differential operator, and B is a "boundary " operator that prescribes values on (possibly only part of) the boundary @\Omega of the underlying bounded domain\Omega 2 IR d . The domain and range spaces can be viewed as certain instances of Sobolev or L 2 spaces such that appropriate trace theorems hold. We can also allow multiple differential, integral, or boundary operators, but we do not want to introduce too much notation at this stage. The goal of this paper is to prove the...

Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

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. The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation. 2000 Mathematics Subject Classification. Primary 41A25, 65N15, 65N35; Secondary 41A05, 41A30, 41A63, 58J40, 65D05. Key words and phrases. Pseudodifferential equation, collocation, zonal kernel, interpolation, approximation order, sphere, positive definite function, radial basis function. x1. Introduction Data fitting and solving differential and integral equations on the sphere are areas of growing interest with applications to physical geodesy, potential theory, oceanography, and meteorology [6,10]. As...

In this paper we provide a framework for studying the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation and approximation on a compact Riemannian manifold. We apply this framework to obtain explicit estimates in cases of the circle and 2-sphere. In addition, we provide a technique for constructing strictly positive definite spherical functions out of radial basis functions, and we use it to make a spherical function that is locally supported.

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the Laplace-Beltrami operator on S n, ω is a non-zero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method

. Positive and conditionally positive definite functions, especially radial basis functions and similar functions for spheres, tori, and even Riemannian manifolds, are of interest because of the their well-known ability to synthesize a good surface fit from scattered data. More recently, positive definite basis functions have been employed to analyze scattered data. The methods used to do this involve constructing multiresolution analyses or multilevel approximations. This paper will discuss recent developments in the synthesis and analysis problems, point out new directions in their investigation, and remark on applications. x1. Introduction Positive definite and conditionally positive definite functions and kernels are used in areas that require fitting a surface to data taken at scattered points in Euclidean space or on some surface, a sphere or torus, say. When the underlying space is Euclidean, radial basis functions (RBFs)--- e.g., Gaussians, multiquadrics, and thin-plate spline...

Governmental purposes notwithstanding any copyright notation thereon. The views conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government. 1 2 For interpolation of smooth functions by smooth kernels having an expansion into eigenfunctions (e.g. on the circle, the sphere, and the torus), good results including error bounds are known, provided that the smoothness of the function is closely related to that of the kernel. The latter fact is usually quantified by the requirement that the function should lie in the “native ” Hilbert space of the kernel, but this assumption rules out the treatment of less smooth functions by smooth kernels. For the approximation of functions from “large ” Sobolev spaces W by functions generated by smooth kernels, this paper shows that one gets at least the known order for interpolation with a nonsmooth kernel that has W as its native space.