Box averaging filter ($G_{\Delta }(r)={\Delta }^{-1}{1}_{\{|r|\le \Delta /2\}}$)

A box averaging filter gives equal weight to all positions within a specified window and zero weight outside it. In one spatial dimension, its normalized kernel and action areThis is a spatial averaging filter expressed as a convolution. In several dimensions, averaging over an axis-aligned rectangular box uses the product of the one-dimensional kernels.

The discrete counterpart averages a fixed number of samples. On an infinite sequence that is zero except for value ten at index zero, a centred five-sample box average produces value two at indices $-2,-1,0,1,2$ and zero elsewhere. The total remains ten, while the isolated peak spreads. On a finite array, boundary extension must be specified; reflection, zero padding and wrapping can give different edge values.^[ref]

Applying the same box average twice is not the same operation as applying it once. The new kernel is the convolution of the two kernels:Indeed, the integral counts the length of overlap of two intervals, divided by ${\Delta }^2$. The resulting support has width $2\Delta$, and the weight decreases away from its centre.

For a Fourier mode $\cos (kx)$, one application multiplies the amplitude by $\sin (k\Delta /2)/(k\Delta /2)$; two applications square this multiplier. The sinc function response explains why a box average smooths a signal without implementing an ideal sharp frequency cutoff. In large eddy simulation, that distinction matters when relating a grid filter to a coarser test filter.