Orthonormal bases of exponentials for the \(n\)-cube
type: Item
instance of: scholarly article
title: Orthonormal bases of exponentials for the \(n\)-cube
zbMATH Open document ID: 0978.42007
DOI: 10.1215/S0012-7094-00-10312-2
author: Jeffrey C. Lagarias
author: James A. Reeds
author: Yang Wang
published in: Duke Mathematical Journal
publication date: 2000-07-24T00:00:00Z
review text: Consider the unit cube \(Q\) in \({\mathbb R}^d\). It is proved here that a set \(\Lambda \subseteq {\mathbb R}^d\) is a tiling set for \(Q\) (that is, the sets \(Q+\lambda\), \(\lambda\in\Lambda\), are non-overlapping and fill the space) if and only if the set \(E_\Lambda = \{\exp{(2\pi i \lambda\cdot x)}: \lambda\in\Lambda\}\) is a complete orthonormal basis in \(L^2(Q)\). This is related to a conjecture of Fuglede which states that for a domain \(\Omega\) in the space of volume 1, such a basis for \(L^2(\Omega)\) exists if and only if \(\Omega\) can be translated so as to tile the space.
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 11K70
Mathematics Subject Classification ID: 47A13
zbMATH DE Number: 1463077
zbMATH Keywords: orthonormal bases
zbMATH Keywords: exponentials
zbMATH Keywords: tiling set
zbMATH Keywords: conjecture of Fuglede
Wikidata QID: Q56699790
reviewed by: Mihail N. Kolountzakis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1463077