Hermitian wavelet

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

<math>\Psi_{n}(t)=(2n)^{-\frac{n}{2}}c_{n}H_{n}\left(\frac{t}{\sqrt{2n}}\right)e^{-\frac{1}{2n}t^{2}}<math>

where <math>H_{n}\left({x}\right)<math> denotes the <math>n^\textrm{th}<math> Hermite polynomial.

The normalisation coefficient <math>c_{n}<math> is given by:

<math>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}<math>

The prefactor <math>C_{\Psi}<math> in the resolution of the identity of the continuous wavelet transform for this wavelet is given by:

<math>C_{\Psi}=\frac{4\pi n}{2n-1}<math>

i.e. Hermitian wavelets are admissible <math>\forall~n>0<math>.

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