Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics

We consider the optimal recovery problem for the solution of the Dirichlet problem for the Laplace equation in the unit $d$-dimensional ball on a sphere of radius $\rho$ from a finite collection of inaccurately specified Fourier coefficients of the solution on a sphere of radius $r$, $0<r<\rho<1$. The methods are required to be exact on certain subspaces of spherical harmonics.