Hermitian wavelet

From Free net encyclopedia

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{n}}\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>.