Wavelet series

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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.

Contents

[edit] Formal definition

A function <math>\psi\in L^2(\mathbb{R})</math> 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 <math>L^2(\mathbb{R})</math> of square integrable functions. The Hilbert basis is constructed as the family of functions <math>\{\psi_{jk}:j,k\in\Z\}</math> by means of dyadic translations and dilations of <math>\psi\,</math>,

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

for integers <math>j,k\in \mathbb{Z}</math>. This family is an orthonormal system if it is orthonormal under the inner product

<math>\langle\psi_{jk},\psi_{lm}\rangle = \delta_{jl}\delta_{km}</math>

where <math>\delta_{jl}\,</math> is the Kronecker delta and <math>\langle f,g\rangle</math> is the standard inner product on <math>L^2(\mathbb{R})</math>:

<math>\langle f,g\rangle = \int_{-\infty}^\infty \overline{f(x)}g(x)dx</math>

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

<math>f(x)=\sum_{j,k=-\infty}^\infty c_{jk} \psi_{jk}(x)</math>

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.

[edit] Wavelet transform

The integral wavelet transform is the integral transform defined as

<math>\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{|a|}}

\int_{-\infty}^\infty \overline{\psi\left(\frac{x-b}{a}\right)}f(x)dx\,</math>

The wavelet coefficients <math>c_{jk}</math> are then given by

<math>c_{jk}= \left[W_\psi f\right](2^{-j}, k2^{-j})</math>

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

[edit] 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 <math>L^2(\mathbb{R})</math>, is the Haar wavelet.

[edit] See also

[edit] References

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

[edit] External links

ja:ウェーブレット変換 fi:Aallokemuunnos

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