Approximating mixed H\"older functions using random samples

Suppose $f:[0,1{]}^2\to \mathbb{R}$ is a $(c,\alpha )$-mixed H"older 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 $$\mathrm{\parallel }f-\tilde{f}{\mathrm{\parallel }}_{L^2}=\mathcal{O}(n^{-\alpha }{\log }^{3\mathrm{/}2}n),$$ with probability at least $1 - n^{2 -c_1}$,wheretheimplicitconstantonlydependsontheconstants$c > 0$and$c_1 > 0$.