Optimal recovery of three times differentiable functions on a convex polytope inscribed in a sphere

Abstract

We consider the problem of global recovery on the class W3(P) of three times differentiable functions which have uniformly bounded third order derivatives in any direction on a d-dimensional convex polytope P inscribed in a sphere and containing its circumcenter. The information I(f) known about each function f∈W3(P) is given by its values and gradients at the vertices of P. The recovery error is measured in the uniform norm on P. We prove the optimality on the class W3(P) of a certain quasi-interpolating recovery method among all non-adaptive global recovery methods which use the information I(f). This method was constructed earlier for the case of a d-dimensional simplex T in the work by the author and T.S. Sorokina in 2011, where its optimality was proved for an analogous class W2(T) of twice differentiable functions.