Wavelet series

Wavelet series

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

Formal definition

A function psiin L^2(mathbb{R}) is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space L^2(mathbb{R}) of square integrable functions. The Hilbert basis is constructed as the family of functions {psi_{jk}:j,kinZ} by means of dyadic translations and dilations of psi,,

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

for integers j,kin mathbb{Z}. This family is an orthonormal system if it is orthonormal under the inner product

langlepsi_{jk},psi_{lm}rangle = delta_{jl}delta_{km}

where delta_{jl}, is the Kronecker delta and langle f,grangle is the standard inner product langle f,grangle = int_{-infty}^infty overline{f(x)}g(x)dx on L^2(mathbb{R}). The requirement of completeness is that every function fin 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

left[W_psi fright](a,b) = frac{1}{sqrt
int_{-infty}^infty overline{psileft(frac{x-b}{a}right)}f(x)dx,

The wavelet coefficients c_{jk} are then given by

c_{jk}= left[W_psi fright](2^{-j}, k2^{-j})

Here, a=2^{-j} is called the binary dilation or dyadic dilation, and b=k2^{-j} is the binary or dyadic position.

General remarks

Unlike the Fourier transform, which is an integral transform in both directions, the wavelet series is an integral transform in one direction, and a series in the other, much like the Fourier series.

The canonical example of an orthonormal wavelet, that is, a wavelet that provides a complete set of basis elements for L^2(mathbb{R}), is the Haar wavelet.

See also


  • Charles K. Chui, An Introduction to Wavelets, (1992), Academic Press, San Diego, ISBN 0121745848

External links

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