We investigate the least-square approximation to scattered data by d-variate periodic trigonometric polynomials. In a special case our algorithm coincides with the well known ACT method (Adaptive weight, Conjugate gradient acceleration, Toeplitz matrices) but avoids the normal equation, a very important fact in finite precision. The iterative approximation of band-limited functions was very successfully applied in a variety of applications. We generalize this method to the sphere S², where S²: = {x Î R³: ||x||₂ = 1}.
The algorithms are based on iterative CG-type methods in combination with fast Fourier transforms for nonequispaced data. We consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data.
Furthermore we present a fast algorithm for the discrete spherical Fourier transform for scattered data on the sphere.
Numerical examples show the efficiency of the new algorithms.