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

TL;DR

This paper demonstrates that by renorming the space of infinitely differentiable functions, weakly tractable uniform approximation becomes feasible using simple Taylor expansion, challenging prior claims of intractability.

Contribution

It introduces a renorming approach that enables weakly tractable approximation of smooth functions with basic function evaluations, utilizing Taylor's expansion.

Findings

Renorming allows weakly tractable approximation.

Simple Taylor-based algorithms are effective.

Results extend to Euclidean ball and L1-norm approximations.

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.