Patch and crossover planar dyadic wavelet sets

Hergenroeder, A. J.; Catlin, Zachary; George, Brandon; Larson, David

doi:10.2140/involve.2008.1.59

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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

DOI: 10.2140/involve.2008.1.59

Abstract

A single dyadic orthonormal wavelet on the plane $ℝ^{2}$ 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 $ℒ^{2} (ℝ^{2})$ . A planar dyadic wavelet set $E$ is a measurable subset of $ℝ^{2}$ with the property that the inverse Fourier transform of the normalized characteristic function $\frac{1}{2 π} χ (E)$ 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 $ℝ^{2}$ . 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

Vol. 1, No. 1, 2008