Smoothing of Radon projections type of data by bivariate polynomials

Abstract

Given information about a function in two variables, consisting of a finite number of Radon projections, we study the problem of smoothing this data by a bivariate polynomial. It turns out that the smoothing problem is closely connected with the interpolation problem. We propose several schemes consisting of sets of parallel chords in the unit disk which ensure uniqueness of the bivariate polynomial having prescribed Radon projections along these chords. Regular schemes play an important role in both interpolation and smoothing of such kind of data. We prove that the existence and uniqueness of the best smoothing polynomial relies on a regularity property of the scheme of chords. Results of some numerical experiments are presented too.