2006/08/08 by Eugen J. Ionaşcu, Eugen J. Ionascu, Yang Wang +2
Computer Science · Engineering · Mathematics · #11H31 #11H70 #42C40 #FOS: Mathematics #General Mathematics (math.GM) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Sparse and Compressive Sensing Techniques #math.GM #msc:11H31 #msc:11H70 #msc:42C40
paper · pdf · doi:10.48550/arxiv.math/0608200
16 pages, no figures
arxiv created 2006/08/08 · openalex publication_date 2006/08/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Given a lattice in Ł∈ \Rd and a matrix A∈\GLd, does there exist a measurable set T such that both \T+α: α∈Ł\ and \AnT: n∈\Z\ are tilings of \Rd? This problem comes directly from the study of wavelets and wavelet sets. Such a T is known to exist if A is expanding. When A is not expanding the problem becomes much more subtle. Speegle \citeSpe03 exhibited examples in which such a T exists for some Ł and nonexpanding A in \R2. In this paper we give a complete solution to this problem in \R2.