打包、平铺、正交性和完整性
Packing, tiling, orthogonality and completeness
摘要：
设$Ω\subseteq {\bf R}^d$是测度为1的开集。如果$D-D$不与$Ω$的指示函数的傅立叶变换的零点相交，并且$D$具有测度1，则开集$D \subseteq {\bf R}^d$被称为$Ω$的"紧正交填充区域“。假设$Λ$是${\bf R}^d$的离散子集。本文的主要贡献是用一种新的方法证明了以下结果（Lagarias，Reeds和Wang用不同的方法证明了，在$Ω$是立方体的情况下，Iosevich和Pedersen用不同的方法证明了）：当在位置$Λ$平移时，$D$瓦片${\bf R}^d$当且仅当指数集$E_Λ= \{\exp 2πi λ\cdot x: λ\inΛ\}$是$L^2(Ω)$的标准正交基。(When $Ω$是${\bf R}^d$中的单位立方体，则它是自身的紧正交填充区域。）在我们的方法中，$E_Λ$的正交性被看作是一个关于"packing“${\bf R}^d$的陈述，带有某个非负函数的平移，此外，我们在$L^2(Ω)$中具有$E_Λ$的完备性当且仅当上述packing实际上是一个平铺。然后，我们制定了平铺条件傅立叶分析语言，并使用它来证明我们的结果。
英文摘要：
Let $Ω\subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $Ω$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator function of $Ω$ and $D$ has measure 1. Suppose that $Λ$ is a discrete subset of ${\bf R}^d$. The main contribution of this paper is a new way of proving the following result (proved by different methods by Lagarias, Reeds and Wang and, in the case of $Ω$ being the cube, by Iosevich and Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $Λ$ if and only if the set of exponentials $E_Λ= \{\exp 2πi λ\cdot x: λ\inΛ\}$ is an orthonormal basis for $L^2(Ω)$. (When $Ω$ is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of itself.) In our approach orthogonality of $E_Λ$ is viewed as a statement about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function and, additionally, we have completeness of $E_Λ$ in $L^2(Ω)$ if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier Analytic language and use this to prove our result.
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。
