= Repeated symmetric filtering can amplify an oscillation
{title2=$H(\omega)=[4+4\cos\omega-2\cos(2\omega)]/6$}
The symmetric filter with weights $(-1,2,4,2,-1)/6$ has unit gain at zero frequency, but its largest gain is $7/6$ at signed frequencies $\pm\pi/3$. Applied $k$ times, it multiplies a <spectral density> by $|H|^{2k}$. Thus repetition amplifies and selects a period-six oscillation instead of merely suppressing short-scale noise. On a two-sided frequency domain both peaks must be retained.
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