  Differential equations may possess coefficients that vary on a spectrum of
scales. Because coefficients are typically multiplicative in real space, they
turn into convolution operators in spectral space, mixing all wavenumbers.
However, in many applications, only the largest scales of the solution are of
interest and so the question turns to whether it is possible to build effective
coarse-scale models of the coefficients in such a manner that the large scales
of the solution are left intact. Here we apply the method of numerical
homogenization to deterministic linear equations to generate sub-grid-scale
models of coefficients at desired frequency cutoffs. We use the Fourier basis
to project, filter and compute correctors for the coefficients. The method is
tested in 1D and 2D scenarios and found to reproduce the coarse scales of the
solution to varying degrees of accuracy depending on the cutoff. We relate this
method to mode-elimination Renormalization Group (RG) and discuss the
connection between accuracy and the cutoff wavenumber. The tradeoff is governed
by a form of the uncertainty principle for convolutions, which states that as
the convolution operator is squeezed in the spectral domain, it broadens in
real space. As a consequence, basis sparsity is a high virtue and the choice of
the basis can be critical.
