2021/09/21 by Marcin Bownik, Bownik, Marcin, Darrin Speegle +1 · 2 citations
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2109.10323
openalex publication_date 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We solve the wavelet set existence problem. That is, we characterize the full-rank lattices Γ⊂ \mathbb Rn and invertible n × n matrices A for which there exists a measurable set W such that \W + γ: γ∈ Γ\ and \Aj(W): j∈ \mathbb Z\ are tilings of \mathbb Rn. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case n = 2. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues λ satisfy |λ| ≥ 1. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is ≥ 1.