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.

1 Formal definition
2 Wavelet transform
3 General remarks
4 See also
5 References

[edit] Formal definition

A function $\psi\in L^2(\mathbb{R})$ 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 $\{\psi_{jk}:j,k\in\Z\}$ by means of dyadic translations and dilations 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

$\langle\psi_{jk},\psi_{lm}\rangle = \delta_{jl}\delta_{km}$

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

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

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.

[edit] Wavelet transform

The integral wavelet transform is the integral transform defined as

$\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^\infty \overline{\psi\left(\frac{x-b}{a}\right)}f(x)dx\,$

The wavelet coefficients $c j k$ are then given by

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

Here, $a = 2 - j$ is called the binary dilation or dyadic dilation, and $b = k 2 - j$ 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 $L^2(\mathbb{R})$ , is the Haar wavelet.