Uniform recovery of high-dimensional $C^r$-functions

TL;DR

This paper investigates the minimal number of function evaluations needed to uniformly approximate high-dimensional functions with bounded derivatives, revealing new bounds that depend on the smoothness and dimension.

Contribution

It establishes new bounds on the number of function values required for uniform approximation of high-dimensional $C^r$-functions, especially when the smoothness order is even.

Findings

Minimal number of evaluations for integral approximation is of order (d/ε)^{d/r}.

Minimal number of evaluations for uniform approximation is of order (d^{r/2}/ε)^{d/r} for even r.

Provides bounds that depend explicitly on dimension, smoothness, and error tolerance.

Abstract

We consider functions on the $d$-dimensional unit cube whose partial derivatives up to order $r$ are bounded by one. It is known that the minimal number of function values that is needed to approximate the integral of such functions up to the error $\varepsilon$ is of order $(d/ \varepsilon)^{d/r}$. Among other things, we show that the minimal number of function values that is needed to approximate such functions in the uniform norm is of order $(d^{r/2} /\varepsilon)^{d/r}$ whenever $r$ is even.