# Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The [itex]n^\textrm{th}[itex] Hermitian wavelet is defined as the [itex]n^\textrm{th}[itex] derivative of a Gaussian:

[itex]\Psi_{n}(t)=(2n)^{-\frac{n}{2}}c_{n}H_{n}\left(\frac{t}{\sqrt{2n}}\right)e^{-\frac{1}{2n}t^{2}}[itex]

where [itex]H_{n}\left({x}\right)[itex] denotes the [itex]n^\textrm{th}[itex] Hermite polynomial.

The normalisation coefficient [itex]c_{n}[itex] is given by:

[itex]c_{n} = \left(n^{\frac{1}{2}-n}\Gamma(n+\frac{1}{2})\right)^{-\frac{1}{2}} = \left(n^{\frac{1}{2}-n}\sqrt{\pi}2^{-n}(2n-1)!!\right)^{-\frac{1}{2}}\quad n\in\mathbb{Z}[itex]

The prefactor [itex]C_{\Psi}[itex] in the resolution of the identity of the continuous wavelet transform for this wavelet is given by:

[itex]C_{\Psi}=\frac{4\pi n}{2n-1}[itex]

i.e. Hermitian wavelets are admissible [itex]\forall~n>0[itex].

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