Optimal recovery of harmonic functions in the ball from an inaccurately given Radon transform

Consider the Hardy space h_2 of functions harmonic in the unit ball Bbb B^{d}subsetBbb R^d, with norm |f|_{h_2}=sup_{0le r<1}left(int_{Bbb S^{d-1}}|f(rphi)|^2,dphiright)^{1/2}. Denote by Bh_2 the set of functions fin h_2 for which |f|_{h_2}le1. par Define the Radon transform as an integration over hyperplanes in Bbb R^d with respect to the standard measure: Rf(theta,s)=int_{xcdot theta}f(x),ds. Let Z be the set of all hyperplanes in Bbb R^d. Suppose that a function gin L^2(Z) which is close to Rf in the L^2 norm, be given. The problem is to reconstruct f(x) from its inaccurate Radon transform g. par An arbitrary mapping m: L^2(Z)to L^2(Bbb B^d) is called a reconstruction method. The value supnolimits_{fin Bh_2, gin L^2(Z), |Rf-g|_{L^2(Z)}<delta}|f-m(g)|_{L^2(Bbb B^d)} is called a reconstruction error. par A reconstruction method based on the expansion of f into a series over the spherical harmonics is given in the article. It is proved that the proposed method has the minimal reconstruction error.