Randomized mixed Hölder function approximation in higher-dimensions

Abstract

The purpose of this paper is to extend the result of arXiv:1810.00823 to mixed Hölder functions on $[0,1{]}^d$ for all $d\ge 1$. In particular, we prove that by sampling an $\alpha$-mixed Hölder function $f:[0,1{]}^d\to \mathbb{R}$ at $\sim \frac{1}{\epsilon }{\left(\log \frac{1}{\epsilon }\right)}^d$ independent uniformly random points from $[0,1{]}^d$, we can construct an approximation $\tilde{f}$ such that $$\Vert f-\tilde{f}{\Vert }_{L^2}\lesssim {\epsilon }^{\alpha }{\left(\log \frac{1}{\epsilon }\right)}^{d-1/2},$$ with high probability.

21 pages, 4 figures