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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.