Approximating mixed Hölder functions using random samples

Abstract

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