Optimal recovery of isotropic classes of twice-differentiable multivariate functions

We consider the class of functions defined on a convex body in Rd, d∈N, whose second derivatives in any direction are uniformly bounded and the class of d-variate functions periodic with respect to a given full-rank lattice L and having uniformly bounded second derivative in any direction. The problem of the optimal algorithm which recovers functions from these classes using their values and values of their gradients at n points (nodes) is considered. We first obtain an estimate for the error of the optimal algorithms with fixed nodes. In the periodic case, for every n sufficiently large, we describe the optimal set of n nodes. When d=2, for certain periodic cases, optimality of the hexagonal arrangement of nodes is shown. For both the periodic case and the non-periodic case we present asymptotic results as n gets large.