Wavelets, tiling, and spectral sets

Abstract

We consider a function φ ∈ L²(ℝ^d) such that {|det(D)|^1/2φ(Dx - λ) : D ∈ script D sign, λ ∈ script T sign} forms an orthogonal basis for L²(ℝ^d), where script D sign ⊂ M<inf>d</inf>(ℝ) and script T sign ⊂ ℝ^d. Such a function φ is called a wavelet with respect to the dilation set script D sign and translation set script T sign. We study the following question: Under what conditions can a script D sign ⊂ M<inf>d</inf>(ℝ) and a script T sign ⊂ ℝ^d be used as, respectively, the dilation set and the translation set of a wavelet? When restricted to wavelets of the form φ = χ̌ Ω, this question has a surprising tie to spectral sets and their spectra.