= Spatial averaging filter
{title2=$\bar f=G_\Delta*f$}
A <spatial averaging filter> maps a spatially varying quantity to a weighted local average. For a translation-invariant kernel $G_\Delta$ of characteristic width $\Delta$,
$$
\bar f(x)=\int_{\mathbb R^d}G_\Delta(r)f(x-r)\,dr,\qquad
\int_{\mathbb R^d}G_\Delta(r)\,dr=1.
$$
This is a <convolution>. The normalization preserves a constant value; the kernel determines how each spatial scale is changed. In <large eddy simulation>, the filter defines the resolved quantities whose evolution is calculated.<A dynamic subgrid-scale eddy viscosity model by Massimo Germano and others (1990)>[§2, equations 3–7: filter convolution and resulting resolved equations]{ref}
For a one-dimensional <box averaging filter>,
$$
\bar f(x)=\frac1\Delta\int_{x-\Delta/2}^{x+\Delta/2}f(y)\,dy.
$$
Integrating $f(x)=\cos(kx)$ explicitly gives
$$
\bar f(x)=\frac{\sin(k\Delta/2)}{k\Delta/2}\cos(kx).
$$
The multiplier is a <sinc function>. A wave with $k\Delta=\pi$ retains amplitude $2/\pi$, while one with $k\Delta=2\pi$ is removed. This filter attenuates different <Fourier modes> differently; it is not an ideal sharp cutoff of all modes above one threshold.
A linear filter does not preserve products: generally $\overline{fg}\ne\bar f\,\bar g$. That lost product information produces the <subgrid-scale stress>. Nor does every filter satisfy $\overline{\bar f}=\bar f$; applying the box average twice produces a different kernel.
With constant width on an unbounded or periodic region, differentiating the <convolution> gives $\partial_j\bar f=\overline{\partial_jf}$ under suitable regularity. A width varying with position or a special near-wall construction can introduce additional terms. Those assumptions must be stated before commuting a filter through a <partial derivative>.
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