Abstract

A single dyadic orthonormal wavelet on the plane R² 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 L²(R²). A planar dyadic wavelet set E is a measurable subset of R² with the property that the inverse Fourier transform of the normalized characteristic function 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 R². 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

Primary: 47A13, 42C40, 42C15

Authors