Universal sampling, quasicrystals and bounded remainder sets
It is shown that simple quasicrystals have a restricted universal sampling property. A discrete set \(\Lambda\) is said to be a set of sampling for \(\text{PW}_S\), the subspace of \(L^2(\mathbb{R}^d)\) of functions frequency supported in \(S\), if there are constants \(0<c\leq C<\infty \) such that for any \(f\in \text{PW}_S\) one has \(c\|f\|^2\leq \sum_{\lambda\in\Lambda} |f(\lambda)|^2\leq C\|f\|^2\). \(\Lambda\) is a set of (stable) interpolation for \(S\) if for any \(\{c_\lambda\}\in\ell^2(\Lambda)\) there is an \(f\in \text{PW}_S\) such that \(f(\lambda)=c_\lambda\) (\(\lambda\in \Lambda\)). A set \(\Lambda\) is called a set of universal sampling if it is a set of sampling for any set \(S\) whose measure is smaller than the lower density \(D^{-}(\Lambda)=\lim\inf_{R\to\infty}\inf_{x\in\mathbb{R}^d} \# (\Lambda\cap B(x;R))/|B(R)|\) of \(S\) and is called a set of universal interpolation if it is a set of stable interpolation for any set of measure larger than the upper density \(D^{+}(\Lambda)=\lim\sup_{R\to\infty}\sup_{x\in\mathbb{R}^d} \# (\Lambda\cap B(x;R))/|B(R)|\). Existence of universal sequences of sampling and of interpolation were proved independently by \textit{A. Olevskiĭ} and \textit{A. Ulanovskii} [Geom. Funct. Anal. 18, No. 3, 1029--1052 (2008; Zbl 1169.42014)] of \textit{B. Matei} and \textit{Y. Meyer} [C. R., Math., Acad. Sci. Paris 346, No. 23--24, 1235--1238 (2008; Zbl 1154.42006)]. The present work establishes conditions for a restricted type of universality that applies to particular sets \(S\), \(|S|=D(\Lambda)\) (when \(D^{+}(\Lambda)=D^{-}(\Lambda)\)). Specifically, the \(S\) to which this restricted universality applies are Riemann measurable, meaning the boundary of \(S\) has measure zero and \(S\) is equidecomposable to a parellelepiped, meaning that \(S\) can be partitioned into finitely many subsets that can be reassembled into a parallelepiped after rigid motions. \(\Lambda\) is a simple quasicrystal if there is a lattice \(\Gamma\) in \(\mathbb{R}^d\times\mathbb{R}\) and an interval \(I=[a,b)\) such that the projections \(p_1(x_1,\dots, x_{n+1})=(x_1,\dots, x_n)\) and \(p_2(x_1,\dots,x_{n+1})=x_{n+1}\), produce \(\Lambda\) from \((\Gamma,I)\) in the sense that \(\Lambda=\Lambda(\Gamma,I)=\{p_1(\gamma):\, \gamma\in\Gamma,\, p_2(\gamma)\in I\}\). In this case \(D(\Lambda)=|I|/\det{\Gamma}\). The main results for \(\Lambda\) a simple quasicrystal are as follows. (1) if \(|I|\in p_2(\Gamma)\) then \(\Lambda\) is a simultaneous set of sampling and interpolation for every Riemann measurable set \(S\) such that \(|S|=D(\Lambda)\) and such that \(S\) is equidecomposable to a parallelepiped with vertices in \(p_1(\Gamma^\ast)\), where \(\Gamma^\ast\) is the dual lattice of \(\Gamma\), using translations by vectors in \(p_1(\Gamma^\ast)\); (2) if \(|I|\notin p_2(\Gamma)\) then there is not Riemann measurable set such that \(\Lambda\) is a simultaneous set of sampling and interpolation for \(\text{PW}_S\) and (3) \(\Lambda(\Gamma,I)\) is a simultaneous set of sampling and interpolation for \(\text{PW}_S\) for every Riemann measurable bounded remainder set \(S\) (a set whose measure is always close to the average value of any \(n\) consecutive shifts of its characteristic function) with \(|S|=|I|\).NEWLINENEWLINEAs an example, \(\lambda(n)=n+n\alpha-[n\alpha]\) is a universal set of sampling and interpolation for \(S\) whenever \(S\) is a finite union of intervals with lengths in \(\alpha\mathbb{Z}+\mathbb{Z}\) that sum to one.
scientific article; zbMATH DE number 6339675
scientific article