Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions

Abstract

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that approximation of infinitely differentiable functions is intractable and suffers from the curse of dimensionality. We show that renorming the space of infinitely differentiable functions in a suitable way allows weakly tractable uniform approximation by using only function values. Moreover, the approximating algorithm is based on a simple application of Taylor's expansion at the center of the unit cube. We discuss also the approximation on the Euclidean ball and the approximation in the $L_1$-norm, which is closely related to the problem of numerical integration.