Optimal Order of Convergence and (In)Tractability of Multivariate Approximation of Smooth Functions

Abstract

We study the approximation problem for C∞ functions f:[0,1]d→ℝ with respect to a Wpm-norm. Here, m=[m,m,…,m], d times, with the norm of the target space defined in terms of up to m partial derivatives with respect to all d variables. The optimal order of convergence is infinite, hence excellent, but the problem is still intractable and suffers from the curse of dimensionality if m≥1. This means that the order of convergence supplies incomplete information concerning the computational difficulty of a problem. For m=0 and p=2, we prove that the problem is not polynomially tractable, but that it is weakly tractable.