Hermitian wavelet

Hermitian wavelets are a family of discrete and continuous wavelets used in the continuous and discrete Hermite wavelet transform. The $$n^\textrm{th}$$ Hermitian wavelet is defined as the $$n^\textrm{th}$$ derivative of a Gaussian distribution for each positive $$n$$: $$\Psi_{n}(x)=(2n)^{-\frac{n}{2}}c_{n}\operatorname{He}_{n}\left(x\right)e^{-\frac{1}{2}x^{2}}, $$where in this case the (probabilist) Hermite polynomial $$\operatorname{He}_{n}(x)$$ can be considered.

The normalization coefficient $$c_{n}$$ is given by$$c_{n} = \left(n^{\frac{1}{2}-n}\Gamma\left(n+\frac{1}{2}\right)\right)^{-\frac{1}{2}} = \left(n^{\frac{1}{2}-n}\sqrt{\pi}2^{-n}(2n-1)!!\right)^{-\frac{1}{2}}\quad n\in\mathbb{N}.$$The function $$\Psi\in L_{\rho, \mu}(-\infty, \infty)$$ is said to be an admissible Hermite wavelet if it satisfies the admissibility relation:

$$C_\Psi = \sum_{n=0}^{\infty}{\frac{\|\hat\Psi (n)\|^2}{\|n\|}} < \infty$$

where $$\hat \Psi (n)$$ is the Hermite transform of $$\Psi$$.

The perfector $$C_{\Psi}$$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula$$C_{\Psi}=\frac{4\pi n}{2n-1}.$$In computer vision and image processing, Gaussian derivative operators of different orders are frequently used as a basis for expressing various types of visual operations; see scale space and N-jet.

Examples
The first three derivatives of the Gaussian function with $$\mu=0,\;\sigma=1$$:$$f(t) = \pi^{-1/4}e^{(-t^2/2)},$$are:$$\begin{align} f'(t) & = -\pi^{-1/4}te^{(-t^2/2)}, \\ f''(t)         & = \pi^{-1/4}(t^2 - 1)e^{(-t^2/2)},\\ f^{(3)}(t) & = \pi^{-1/4}(3t - t^3)e^{(-t^2/2)}, \end{align}$$and their $$L^2$$ norms $$\lVert f' \rVert=\sqrt{2}/2, \lVert f'' \rVert=\sqrt{3}/2, \lVert f^{(3)} \rVert= \sqrt{30}/4$$.

Normalizing the derivatives yields three Hermitian wavelets:$$\begin{align} \Psi_{1}(t) &= \sqrt{2}\pi^{-1/4}te^{(-t^2/2)},\\ \Psi_{2}(t) &=\frac{2}{3}\sqrt{3}\pi^{-1/4}(1-t^2)e^{(-t^2/2)},\\ \Psi_{3}(t) &= \frac{2}{15}\sqrt{30}\pi^{-1/4}(t^3 - 3t)e^{(-t^2/2)}. \end{align}$$