Patch and crossover planar dyadic wavelet sets

Abstract

A single dyadic orthonormal wavelet on the plane ${ℝ}^2$ is a measurable square integrable function $\psi \left(x,y\right)$ whose images under translation along the coordinate axes followed by dilation by positive and negative integral powers of 2 generate an orthonormal basis for ${\mathcal{ℒ}}^2\left({ℝ}^2\right)$. 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 $\frac{1}{2\pi }\chi \left(E\right)$ 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.