= Evenization of a nonnegative bandlimited filter
{title2=$w(t)=C f(t/2)f(-t/2)$}
For a nonzero nonnegative integrable <bandlimited function> $f$ with frequency support in $[-\Delta,\Delta]$, the displayed construction is even, nonnegative and normalizable. Each factor has half-width frequency support, so their product retains width $\Delta$. Boundedness gives $\int_T^\infty w(t)dt\leq2C\|f\|_\infty\int_{T/2}^\infty f(u)du$. Thus a positive-time tail estimate becomes a two-sided estimate after evenization, with a rescaled decay constant. A simple average with $f(-t)$ would not establish this from an uncontrolled negative tail.
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