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A bandlimited function has Fourier transform supported in a bounded frequency interval. An integrable bandlimited function has a bounded continuous representative when its Fourier transform is integrable, and that representative extends analytically using the compact-frequency integral. Scaling arguments scales its frequency support inversely, while multiplying two such functions adds their support intervals through convolution.

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 ${\int }_T^{\infty }w(t)dt\le 2C\Vert f{\Vert }_{\infty }{\int }_{T/2}^{\infty }f(u)du$. 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.