Keiner, Jens ; Kunis, Stefan ; Potts, Daniel : Efficient reconstruction of functions on the sphere from scattered data

Author(s):

Keiner, Jens
Kunis, Stefan
Potts, Daniel

Title:

Efficient reconstruction of functions on the sphere from scattered data

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 22, 2006

Mathematics Subject Classification:

65T50			[ Discrete and fast Fourier transforms ]
33C55			[ Spherical harmonics ]
65F10			[ Iterative methods for linear systems ]
65T40			[ Trigonometric approximation and interpolation ]

Abstract:

Recently, fast and reliable algorithms for the evaluation of spherical harmonic expansions have been developed. The corresponding sampling problem is the computation of Fourier coefficients of a function from sampled values at scattered nodes. We consider a least squares approximation to and an interpolation of given data. Our main result is that the rate of convergence of the two proposed iterative schemes depends only on the mesh norm and the separation distance of the nodes. In conjunction with the nonequispaced FFT on the sphere, the reconstruction of $N^2$ Fourier coefficients from $M$ reasonably distributed samples is shown to take $\cO(N^2 \log^2 N+M)$ floating point operations. Numerical results support our theoretical findings.

Keywords:

approximation by spherical harmonics, scattered data interpolation, iterative methods, nonequispaced FFT on the sphere

Language:

English

Publication time:

11 / 2006