Approximating mixed Hölder functions using random samples

Abstract

Suppose $f:[0,1{]}^2\to \mathbb{R}$ is a $(c,\alpha )$-mixed Hölder function that we sample at $l$ points $X_1,\dots ,X_l$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_1),\dots ,f(X_l)$ be given. If $l\ge c_{1}n{\log }^{2}n$, then we can compute an approximation $\tilde{f}$ such that $$\Vert f-\tilde{f}{\Vert }_{L^2}=\mathcal{O}(n^{-\alpha }{\log }^{3/2}n),$$ with probability at least $1-n^{2-c_1}$, where the implicit constant only depends on the constants $c>0$ and $c_1>0$.

15 pages, 3 figures