Shannon wavelet

Example

Write $\operatorname{sinc}(t) = \sin(\pi t)/(\pi t)$ for $t \ne 0$ and $\operatorname{sinc}(0) = 1$. Use the Fourier transform $\hat f(\omega) = \int_{\R} f(t) e^{-i\omega t} \, dt$. The real Shannon wavelet is $$\psi(t) = 2\operatorname{sinc}(2t) - \operatorname{sinc}(t) = \operatorname{sinc}\left(\frac{t}{2}\right) \cos\left(\frac{3\pi t}{2}\right).$$ Its Fourier transform is the indicator function of the set $\set{\omega : \pi < \abs{\omega} \le 2\pi}$, up to a set of measure zero.

Check. The transform of $\operatorname{sinc}$ is the indicator of $[-\pi, \pi]$. The transform of $2\operatorname{sinc}(2t)$ is the indicator of $[-2\pi, 2\pi]$. Their difference is the indicator of the two bands. The second formula follows from $\sin(2\pi t) - \sin(\pi t) = 2\cos(3\pi t/2)\sin(\pi t/2)$.