Wavelet Transform Article Index for
Wavelet 		Website Links For
Series 		 

Information About

Wavelet Transform
  CATEGORIES ABOUT WAVELET SERIES
   
wavelets
   
functional analysis
   
SHOPPER'S DELIGHT
 
Shop For:  
Department
Price Minimum:
Price Maximum:
      




FORMAL DEFINITION

A function \psi\in L^2(\mathbb{R}) is called an orthonormal wavelet if it can be used to define a Translation s and Dilation s of \psi\,,

:\psi_{jk}(x) = 2^{j/2} \psi(2^jx-k)\,

for integers j,k\in \mathbb{Z}. This family is an orthonormal system if it is orthonormal under the Inner Product

:\langle\psi_{jk},\psi_{lm} angle = \delta_{jl}\delta_{km}

where \delta_{jl}\, is the Kronecker Delta and \langle f,g angle is the standard inner product on L^2(\mathbb{R}):

:\langle f,g angle = \int_{-\infty}^\infty \overline{f(x)}g(x)dx

The requirement of completeness is that every function f\in L^2(\mathbb{R}) may be expanded in the basis as

:f(x)=\sum_{j,k=-\infty}^\infty c_{jk} \psi_{jk}(x)

with convergence of the series understood to be Convergence In The Norm . Such a representation of a function ''f'' is known as a wavelet series. This implies that an orthonormal wavelet is Self-dual .


WAVELET TRANSFORM

The integral wavelet transform is the Integral Transform defined as