A Matching Pursuit for the Sphere Based on the Poisson Semigroup

An iterative greedy sampling and reconstruction algorithm for square-integrable functions on the two-sphere is presented. The algorithm performs a decomposition based on reproducing kernel spaces related to the Poisson semigroup. This decomposition starts from the harmonic extension of the initial function to the ball. In each iteration step, the respective input function is sampled in the ball and a multiple of a suitably chosen kernel function is subtracted. We discuss the rate of convergence for the reconstruction of functions that have harmonic extensions with finite gradient norm in the ball.