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Wavelets with composite dilations

Author(s): Kanghui Guo; Demetrio Labate; Wang-Q Lim; Guido Weiss; Edward Wilson
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78-87.
MSC (2000): Primary 42C15, 42C40
Posted: August 3, 2004
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Abstract: A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2({\mathbb R}^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are shear matrices (all eigenvalues are one), while the members of $A$ are matrices expanding or contracting on a proper subspace of ${\mathbb R}^n$. These wavelets are of interest in applications because of their tendency to produce ``long, narrow'' window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

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Additional Information:

Kanghui Guo
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804
Email: kag026f@smsu.edu

Demetrio Labate
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: dlabate@math.ncsu.edu

Wang-Q Lim
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email: wangQ@math.wustl.edu

Guido Weiss
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email: guido@math.wustl.edu

Edward Wilson
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email: enwilson@math.wustl.edu

DOI: 10.1090/S1079-6762-04-00132-5
PII: S 1079-6762(04)00132-5
Keywords: Affine systems, frames, multiresolution analysis (MRA), multiwavelets, wavelets
Received by editor(s): February 23, 2004
Received by editor(s) in revised form: April 13, 2004
Posted: August 3, 2004
Additional Notes: The fourth author was supported in part by a SW Bell Grant.
Communicated by: Boris Hasselblatt
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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