Efficient Reconstruction of Functions on the Sphere from Scattered Data

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 and an interpolation of the 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 N² Fourier coefficients from M reasonably distributed samples is shown to take O(N² log² N + M) floating point operations. Numerical results support our theoretical findings.