Approximating mixed Holder functions using random samples

Suppose $f : [0,1]^2 rightarrow mathbb{R}$ is a $(c,alpha)$-mixed Holder 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 $$ |f - tilde{f} |_{L^2} = mathcal{O}(n^{-alpha} log^{3/2} n), $$ with probability at least $1 - n^{2 -c_1}$, where the implicit constant only depends on the constants $c > 0$ and $c_1 > 0$.