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Wavelets with composite dilations
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## Wavelets with composite dilations

### Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss, and Edward Wilson

**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.

*Copyright 2004 American Mathematical Society*

The copyright for this article reverts to public domain after 28 years from publication.

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#### Article Info

- ERA Amer. Math. Soc.
**10** (2004), pp. 78-87
- Publisher Identifier: S 1079-6762(04)00132-5
- 2000
*Mathematics Subject Classification*. Primary 42C15, 42C40
*Key words and phrases*. Affine systems, frames, multiresolution analysis (MRA),
multiwavelets, wavelets
- Received by editors February 23, 2004
- Received by editors in revised form April 13, 2004
- Posted on August 3, 2004
- Communicated by Boris Hasselblatt
- Comments (When Available)

**Kanghui Guo**

Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804

*E-mail address:* `kag026f@smsu.edu`

**Demetrio Labate**

Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

*E-mail address:* `dlabate@math.ncsu.edu`

**Wang-Q Lim**

Department of Mathematics, Washington University, St. Louis, Missouri 63130

*E-mail address:* `wangQ@math.wustl.edu}`

**Guido Weiss**

Department of Mathematics, Washington University, St. Louis, Missouri 63130

*E-mail address:* `guido@math.wustl.edu}`

**Edward Wilson**

Department of Mathematics, Washington University, St. Louis, Missouri 63130

*E-mail address:* `enwilson@math.wustl.edu}`

The fourth author was supported in part by a SW Bell Grant.

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