Multiscale Analysis in Sobolev Spaces on the Sphere

Authors:
Q. T. Le Gia;I. H. Sloan;H. Wendland
Affiliations:
qlegia@unsw.edu.au and i.sloan@unsw.edu.au;-;holger.wendland@maths.ox.ac.uk
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

Citing 5
Cited 1

Norm estimates for the inverse of a general class of scattered-data radial-function interpolation matrices

Journal of Approximation Theory
Multistep scattered data interpolation using compactly supported radial basis functions

Journal of Computational and Applied Mathematics - Special issue on scattered data
Compactly supported radial functions and the Strang-Fix condition

Applied Mathematics and Computation
Error estimates for interpolation by compactly supported radial basis functions of minimal degree

Journal of Approximation Theory
Continuous and discrete least-squares approximation by radial basis functions on spheres

Journal of Approximation Theory

Quasi-uniformity of minimal weighted energy points on compact metric spaces

Journal of Complexity

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Abstract

We consider a multiscale approximation scheme at scattered sites for functions in Sobolev spaces on the unit sphere $\mathbb{S}^n$. The approximation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to $\mathbb{S}^n$. A convergence theorem for the scheme is proved, and the condition number of the linear system is shown to stay bounded by a constant from level to level, thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points.