Nonperiodic sampling and the local three squares theorem
type: Item
instance of: scholarly article
title: Nonperiodic sampling and the local three squares theorem
zbMATH Open document ID: 1012.42003
DOI: 10.1007/BF02384491
author: Christopher Heil
author: Karlheinz Gröchenig
author: David F. Walnut
published in: Arkiv för Matematik
publication date: 2003-03-25T00:00:00Z
review text: The local three squares problem concerns the question whether functions defined on a cube in \(\mathbb{R}^d\) are determined from their averages on certain smaller cubes. The authors prove a quite general result, namely: fix the dimension \(d\), consider positive numbers \(r_1, \dots, r_{d+1}\), and take any \(R\geq \sum_{i=1}^{d+1}r_i\). Then any function \(f\in L^2([-R,R]^d)\) is determined by its averages over cubes of side \(r_i\) that are completely contained in \([-R,R]^d\) and have edges parallel to the coordinate axes, if and only if \(r_i/r_j\) is irrational for \(i\neq j\). The main technical tool is the sampling theory for bandlimited functions.
Mathematics Subject Classification ID: 42A65
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42A70
Mathematics Subject Classification ID: 94C10
zbMATH DE Number: 1887456
zbMATH Keywords: exact sequences
zbMATH Keywords: irregular sampling
zbMATH Keywords: nonperiodic sampling
zbMATH Keywords: Pompeiu problems
zbMATH Keywords: three squares theorem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02384491
OpenAlex ID: W2092778077
description: scientific article; zbMATH DE number 1887456