Ricker wavelet

Example

Fix a width $\sigma > 0$. The Ricker wavelet, also called the Mexican hat wavelet, is $$\psi(t) = \frac{2}{\sqrt{3\sigma} \, \pi^{1/4}} \left(1 - \frac{t^2}{\sigma^2}\right) e^{-t^2/(2\sigma^2)}, \qquad t \in \R.$$ Up to a negative constant factor, it is the second derivative of the Gaussian $e^{-t^2/(2\sigma^2)}$.

Check of the zero mean, with $\sigma = 1$. Since $\int_{\R} e^{-t^2/2} \, dt = \sqrt{2\pi}$ and $\int_{\R} t^2 e^{-t^2/2} \, dt = \sqrt{2\pi}$, the integral of $(1 - t^2) e^{-t^2/2}$ is $0$. Check of the norm, with $\sigma = 1$. The integral of $(1 - t^2)^2 e^{-t^2}$ is $\sqrt{\pi}\,(1 - 1 + 3/4) = 3\sqrt{\pi}/4$, and multiplying by $4/(3\sqrt{\pi})$ gives $\norm{\psi}_2^2 = 1$. The general case follows by the substitution $t = \sigma s$.