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A wavelet is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300301.png" /> that yields a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300302.png" /> by means of translations and dyadic dilations of itself, i.e.,
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If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300303.png" /></td> </tr></table>
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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300304.png" /> (cf. also [[Wavelet analysis|Wavelet analysis]]). Such a decomposition is called the discrete wavelet transform.
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A wavelet is a function $\psi \in L ^ { 2 } ( \mathbf{R} )$ that yields a basis in $L ^ { 2 } ( \mathbf{R} )$ by means of translations and dyadic dilations of itself, i.e.,
  
In 1988, the Belgian mathematician I. Daubechies constructed [[#References|[a2]]] a class of wavelet functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300306.png" />, that satisfy some special properties. First of all, the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300308.png" />, is an [[Orthonormal system|orthonormal system]] for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d1300309.png" />. Furthermore, each wavelet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003010.png" /> is compactly supported (cf. also [[Function of compact support|Function of compact support]]). Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003011.png" />. The index number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003012.png" /> is also related to the number of vanishing moments, i.e.,
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\begin{equation*} f ( x ) = \sum _ { j = - \infty } ^ { \infty } \sum _ { k = - \infty } ^ { \infty } a _ { j , k } \psi ( 2 ^ { j } x - k ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003013.png" /></td> </tr></table>
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for all $f \in L ^ { 2 } ( \mathbf{R} )$ (cf. also [[Wavelet analysis|Wavelet analysis]]). Such a decomposition is called the discrete wavelet transform.
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In 1988, the Belgian mathematician I. Daubechies constructed [[#References|[a2]]] a class of wavelet functions $\psi _ { N }$, $N \in \mathbf{N} \backslash \{ 0 \}$, that satisfy some special properties. First of all, the collection $\psi _ { N } ( x - k )$, $k \in \mathbf{Z}$, is an [[Orthonormal system|orthonormal system]] for fixed $N \in \mathbf{N} \backslash \{ 0 \}$. Furthermore, each wavelet $\psi _ { N }$ is compactly supported (cf. also [[Function of compact support|Function of compact support]]). Moreover, $\operatorname { supp } ( \psi _ { N } ) = [ 0,2 N - 1 ]$. The index number $N$ is also related to the number of vanishing moments, i.e.,
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\begin{equation*} \int _ { - \infty } ^ { \infty } x ^ { k } \psi _ { N } ( x ) d x = 0,0 \leq k \leq N. \end{equation*}
  
 
A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact,
 
A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003014.png" /></td> </tr></table>
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\begin{equation*} \exists \lambda &gt; 0 \forall N \in \mathbf{N} , N &gt; 2 : \psi _ { N } \in C ^ { \lambda N }. \end{equation*}
  
For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003015.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003016.png" />.
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For large $N$ one has $\lambda \approx 0.2$.
  
The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003017.png" />, which is in fact the Haar wavelet [[#References|[a3]]]. Satisfying symmetry conditions cannot go together with all other properties of the Daubechies wavelets.
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The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for $\Psi_1$, which is in fact the Haar wavelet [[#References|[a3]]]. Satisfying symmetry conditions cannot go together with all other properties of the Daubechies wavelets.
  
 
The Daubechies wavelets can also be used for the continuous wavelet transform, i.e.
 
The Daubechies wavelets can also be used for the continuous wavelet transform, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003018.png" /></td> </tr></table>
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\begin{equation*} W _ { \psi } [ f ] ( a , b ) = \frac { 1 } { \sqrt { a } } \int _ { - \infty } ^ { \infty } f ( x ) \psi \overline{\left( \frac { x - b } { a } \right)} d x, \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003021.png" />. The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003023.png" /> denote scale and translation/position of the transform. A stable reconstruction formula exists for the continuous wavelet transform if and only if the following admissibility condition holds:
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for $f \in L ^ { 2 } ( \mathbf{R} )$, $a \in \mathbf{R} ^ { + }$ and $b \in \mathbf{R}$. The parameters $a$ and $b$ denote scale and translation/position of the transform. A stable reconstruction formula exists for the continuous wavelet transform if and only if the following admissibility condition holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003024.png" /></td> </tr></table>
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\begin{equation*} 0 &lt; C _ { \psi } = 2 \pi \int _ { 0 } ^ { \infty } \frac { \left| \widehat { \psi } ( a \omega ) \right| ^ { 2 } } { a } d a &lt; \infty , \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003025.png" /> denotes the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003026.png" />. The reconstruction formula reads:
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where $\widehat { \psi }$ denotes the [[Fourier transform|Fourier transform]] of $\psi$. The reconstruction formula reads:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003027.png" /></td> </tr></table>
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\begin{equation*} f ( x ) = \frac { 1 } { C _ { \psi } } \int _ { 0 } ^ { \infty } \int _ { - \infty } ^ { \infty } W _ { \psi } [ f ] ( a , b ) \psi ( \frac { x - b } { a } ) d b \frac { d a } { a \sqrt { a } }. \end{equation*}
  
This result holds weakly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003028.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130030/d13003030.png" />, this results also holds pointwise.
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This result holds weakly in $L ^ { 2 } ( \mathbf{R} )$. For $f \in L ^ { 1 } ( \mathbf{R} ) \cap L ^ { 2 } ( \mathbf{R} )$ and $\hat { f } \in L ^ { 1 } ( \mathbf{R} )$, this results also holds pointwise.
  
 
All Daubechies wavelets satisfy the admissibility condition and thus guarantee a stable reconstruction.
 
All Daubechies wavelets satisfy the admissibility condition and thus guarantee a stable reconstruction.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Daubechies,  "Ten lectures on wavelets" , SIAM  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Daubechies,  "Orthonormal bases of compactly supported wavelets"  ''Commun. Pure Appl. Math.'' , '''41'''  (1988)  pp. 909–996</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Haar,  "Zur theorie der orthogonalen Funktionensysteme"  ''Math. Ann.'' , '''69'''  (1910)  pp. 331–371</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Daubechies,  "Ten lectures on wavelets" , SIAM  (1992)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Daubechies,  "Orthonormal bases of compactly supported wavelets"  ''Commun. Pure Appl. Math.'' , '''41'''  (1988)  pp. 909–996</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Haar,  "Zur theorie der orthogonalen Funktionensysteme"  ''Math. Ann.'' , '''69'''  (1910)  pp. 331–371</td></tr></table>

Revision as of 15:30, 1 July 2020

A wavelet is a function $\psi \in L ^ { 2 } ( \mathbf{R} )$ that yields a basis in $L ^ { 2 } ( \mathbf{R} )$ by means of translations and dyadic dilations of itself, i.e.,

\begin{equation*} f ( x ) = \sum _ { j = - \infty } ^ { \infty } \sum _ { k = - \infty } ^ { \infty } a _ { j , k } \psi ( 2 ^ { j } x - k ), \end{equation*}

for all $f \in L ^ { 2 } ( \mathbf{R} )$ (cf. also Wavelet analysis). Such a decomposition is called the discrete wavelet transform.

In 1988, the Belgian mathematician I. Daubechies constructed [a2] a class of wavelet functions $\psi _ { N }$, $N \in \mathbf{N} \backslash \{ 0 \}$, that satisfy some special properties. First of all, the collection $\psi _ { N } ( x - k )$, $k \in \mathbf{Z}$, is an orthonormal system for fixed $N \in \mathbf{N} \backslash \{ 0 \}$. Furthermore, each wavelet $\psi _ { N }$ is compactly supported (cf. also Function of compact support). Moreover, $\operatorname { supp } ( \psi _ { N } ) = [ 0,2 N - 1 ]$. The index number $N$ is also related to the number of vanishing moments, i.e.,

\begin{equation*} \int _ { - \infty } ^ { \infty } x ^ { k } \psi _ { N } ( x ) d x = 0,0 \leq k \leq N. \end{equation*}

A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact,

\begin{equation*} \exists \lambda > 0 \forall N \in \mathbf{N} , N > 2 : \psi _ { N } \in C ^ { \lambda N }. \end{equation*}

For large $N$ one has $\lambda \approx 0.2$.

The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for $\Psi_1$, which is in fact the Haar wavelet [a3]. Satisfying symmetry conditions cannot go together with all other properties of the Daubechies wavelets.

The Daubechies wavelets can also be used for the continuous wavelet transform, i.e.

\begin{equation*} W _ { \psi } [ f ] ( a , b ) = \frac { 1 } { \sqrt { a } } \int _ { - \infty } ^ { \infty } f ( x ) \psi \overline{\left( \frac { x - b } { a } \right)} d x, \end{equation*}

for $f \in L ^ { 2 } ( \mathbf{R} )$, $a \in \mathbf{R} ^ { + }$ and $b \in \mathbf{R}$. The parameters $a$ and $b$ denote scale and translation/position of the transform. A stable reconstruction formula exists for the continuous wavelet transform if and only if the following admissibility condition holds:

\begin{equation*} 0 < C _ { \psi } = 2 \pi \int _ { 0 } ^ { \infty } \frac { \left| \widehat { \psi } ( a \omega ) \right| ^ { 2 } } { a } d a < \infty , \end{equation*}

where $\widehat { \psi }$ denotes the Fourier transform of $\psi$. The reconstruction formula reads:

\begin{equation*} f ( x ) = \frac { 1 } { C _ { \psi } } \int _ { 0 } ^ { \infty } \int _ { - \infty } ^ { \infty } W _ { \psi } [ f ] ( a , b ) \psi ( \frac { x - b } { a } ) d b \frac { d a } { a \sqrt { a } }. \end{equation*}

This result holds weakly in $L ^ { 2 } ( \mathbf{R} )$. For $f \in L ^ { 1 } ( \mathbf{R} ) \cap L ^ { 2 } ( \mathbf{R} )$ and $\hat { f } \in L ^ { 1 } ( \mathbf{R} )$, this results also holds pointwise.

All Daubechies wavelets satisfy the admissibility condition and thus guarantee a stable reconstruction.

References

[a1] I. Daubechies, "Ten lectures on wavelets" , SIAM (1992)
[a2] I. Daubechies, "Orthonormal bases of compactly supported wavelets" Commun. Pure Appl. Math. , 41 (1988) pp. 909–996
[a3] A. Haar, "Zur theorie der orthogonalen Funktionensysteme" Math. Ann. , 69 (1910) pp. 331–371
How to Cite This Entry:
Daubechies wavelets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Daubechies_wavelets&oldid=49909
This article was adapted from an original article by P.J. Oonincx (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article