An optimal multivariate spline method of recovery of twice differentiable functions

Abstract

A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R d .