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

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that the 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 about the center of the unit cube. We discuss also the approximation on the Euclidean ball and the approximation in the L1L1-norm.