Daubechies wavelets
A wavelet is a function that yields a basis in
by means of translations and dyadic dilations of itself, i.e.,
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for all (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 ,
, that satisfy some special properties. First of all, the collection
,
, is an orthonormal system for fixed
. Furthermore, each wavelet
is compactly supported (cf. also Function of compact support). Moreover,
. The index number
is also related to the number of vanishing moments, i.e.,
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A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact,
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For large one has
.
The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for , 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.
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for ,
and
. The parameters
and
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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where denotes the Fourier transform of
. The reconstruction formula reads:
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This result holds weakly in . For
and
, 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 |
Daubechies wavelets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Daubechies_wavelets&oldid=49909