Vol. 1, No. 1, 2008

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ISSN: 1944-4184 (e-only)
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Patch and crossover planar dyadic wavelet sets

A. J. Hergenroeder, Zachary Catlin, Brandon George and David R. Larson

Vol. 1 (2008), No. 1, 59–90
Abstract

A single dyadic orthonormal wavelet on the plane 2 is a measurable square integrable function ψ(x,y) whose images under translation along the coordinate axes followed by dilation by positive and negative integral powers of 2 generate an orthonormal basis for 2(2). A planar dyadic wavelet set E is a measurable subset of 2 with the property that the inverse Fourier transform of the normalized characteristic function 1 2πχ(E) of E is a single dyadic orthonormal wavelet. While constructive characterizations are known, no algorithm is known for constructing all of them. The purpose of this paper is to construct two new distinct uncountably infinite families of dyadic orthonormal wavelet sets in 2. We call these the crossover and patch families. Concrete algorithms are given for both constructions.

Keywords
wavelet, wavelet set, patch, crossover, congruence
Mathematical Subject Classification 2000
Primary: 47A13, 42C40, 42C15
Milestones
Received: 26 November 2007
Accepted: 12 December 2007
Published: 28 February 2008

Communicated by Kenneth S. Berenhaut
Authors
A. J. Hergenroeder
Department of Mathematics
Davidson College
Box 5677
Davidson, NC 28035
United States
Zachary Catlin
Department of Mathematics
Purdue University
West Lafayette, IN 47907
United States
Brandon George
Department of Mathematics
University of Tennessee
Knoxville, TN 37996
United States
David R. Larson
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
United States
http://www.math.tamu.edu/~larson