Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The ${\displaystyle n^{\textrm {th))}$ Hermitian wavelet is defined as the ${\displaystyle n^{\textrm {th))}$ derivative of a Gaussian distribution, for each positive ${\displaystyle n}$:^[1]

$${\displaystyle \Psi _{n}(t)=(2n)^{-{\frac {n}{2))}c_{n}\operatorname {He} _{n}\left(t\right)e^{-{\frac {1}{2))t^{2)),}$$

${\displaystyle \operatorname {He} _{n}(x)}$

${\displaystyle n^{\textrm {th))}$

The normalization coefficient ${\displaystyle c_{n))$ is given by,

$${\displaystyle c_{n}=\left(n^((\frac {1}{2))-n}\Gamma \left(n+{\frac {1}{2))\right)\right)^{-{\frac {1}{2))}=\left(n^((\frac {1}{2))-n}{\sqrt {\pi ))2^{-n}(2n-1)!!\right)^{-{\frac {1}{2))}\quad n\in \mathbb {N} .}$$

${\displaystyle C_{\Psi ))$

$${\displaystyle C_{\Psi }={\frac {4\pi n}{2n-1))}$$

Examples

The first three derivatives of the Gaussian function with ${\displaystyle \mu =0,\;\sigma =1}$,

$${\displaystyle f(t)=\pi ^{-1/4}e^{(-t^{2}/2)},}$$

$${\displaystyle {\begin{aligned}f'(t)&=-\pi ^{-1/4}te^{(-t^{2}/2)},\\f''(t)&=\pi ^{-1/4}(t^{2}-1)e^{(-t^{2}/2)},\\f^{(3)}(t)&=\pi ^{-1/4}(3t-t^{3})e^{(-t^{2}/2)},\end{aligned))}$$

${\displaystyle L^{2))$

${\displaystyle ||f'||={\sqrt {2))/2,||f''||={\sqrt {3))/2,||f^{(3)}||={\sqrt {30))/4}$

Normalizing the derivatives yields three Hermitian wavelets:

$${\displaystyle {\begin{aligned}\Psi _{1}(t)&={\sqrt {2))\pi ^{-1/4}te^{(-t^{2}/2)},\\\Psi _{2}(t)&={\frac {2}{3)){\sqrt {3))\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)},\\\Psi _{3}(t)&={\frac {2}{15)){\sqrt {30))\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}.\end{aligned))}$$