Equispaced Fourier Matérn matrix error bound

Let the points $x_1,\dots,x_N$ be independently and identically distributed from a bounded probability density function $\rho$ supported on $D=[0,1]^d$. Let the Matérn kernel with parameters $\nu$ and $\ell$ be approximated by equispaced Fourier modes as in Theorem~, with $h$ and $m$ chosen so that the aliasing error is negligible compared to the truncation error.

Equispaced Fourier Matérn matrix error bound. With high probability as $N\to\infty$,

$$\|\tilde K-K\|_F\leq N\tilde{\varepsilon},\qquad \tilde{\varepsilon}=\frac{\tilde{c}_{d,\nu,\rho}}{\ell^{2\nu}(hm)^{2\nu+d/2}},$$

for some constant $\tilde{c}_{d,\nu,\rho}$ independent of $N$, $\ell$, $h$, and $m$.

This heuristic predicts a faster Frobenius-norm convergence rate for less-smooth Matérn kernels than the general uniform-entrywise bound, whose truncation error decays algebraically like $m^{-2\nu}$. It is intended for the iid random-data setting and is not established as a rigorous theorem in the supplied text.