Universal sampling, quasicrystals and bounded remainder sets
type: Item
instance of: scholarly article
title: Universal sampling, quasicrystals and bounded remainder sets
zbMATH Open document ID: 1306.42054
author: Sigrid Grepstad
author: Nir Lev
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2014-09-04T00:00:00Z
Mathematics Subject Classification ID: 42C30
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 94A20
zbMATH DE Number: 6339675
zbMATH Keywords: universal sampling
zbMATH Keywords: universal interpolation
zbMATH Keywords: quasicrystal
zbMATH Keywords: Riemann measurable set
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.crma.2014.05.006
OpenAlex ID: W2012707004
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cites work: Q4527706
DOI: 10.1016/J.CRMA.2014.05.006
review text: 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 \(\Gamm…
description: scientific article; zbMATH DE number 6339675