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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.
## 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 $psi,$,## Wavelet transform

The integral wavelet transform is the integral transform defined as## 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.## See also

## References

## External links

- $psi\_\{jk\}(x)\; =\; 2^\{j/2\}\; psi(2^jx-k),$

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.

- $left[W\_psi\; fright](a,b)\; =\; frac\{1\}\{sqrt$
>}

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.

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.

- Continuous wavelet transform
- Discrete wavelet transform
- Complex wavelet transform
- Dual wavelet
- Multiresolution analysis
- JPEG 2000, a wavelet-based image compression standard
- Some people generate spectrograms using wavelets, called scalograms. Other people generate spectrograms using a short-time Fourier transform
- Chirplet transform
- Time-frequency representation

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

- Robi Polikar The Wavelet Tutorial. .

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Last updated on Monday September 22, 2008 at 05:45:52 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 22, 2008 at 05:45:52 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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