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