R^2上的平移和乘性同时拼接与小波集
Simultaneous Translational and Multiplicative Tiling and Wavelet Sets in R^2
摘要：
同时平铺几个不同的平移集已经研究得相当广泛，特别是与斯坦豪斯问题。近年来对正交小波的研究，特别是对任意伸缩矩阵的研究，导致了对矩阵幂乘的研究。在本文中，我们考虑以下同时平铺问题：给定$Ł\in \R^d$中的一个格和一个矩阵$A\in\GLd$，是否存在一个可测集$T$使得$\{T+α: α\inŁ\}$和$\{A^nT: n\in\Z\}$都是$\R^d$的平铺？这个问题直接来源于小波和小波集的研究。如果$A$正在扩展，则已知存在这样的$T$。当$A$不扩展时，问题变得更加微妙。Speegle \cite{Spe03}展示了这样的例子，其中对于某些$Ł$和$\R^2$中的非扩增$A$存在这样的$T$。在本文中，我们给出了一个完整的解决方案，这个问题在$\R^2$。
英文摘要：
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 $Ł\in \R^d$ and a matrix $A\in\GLd$, does there exist a measurable set $T$ such that both $\{T+α: α\inŁ\}$ and $\{A^nT: n\in\Z\}$ are tilings of $\R^d$? 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 \cite{Spe03} exhibited examples in which such a $T$ exists for some $Ł$ and nonexpanding $A$ in $\R^2$. In this paper we give a complete solution to this problem in $\R^2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
