Discrete analogues in harmonic analysis: spherical averages
type: Item
instance of: scholarly article
title: Discrete analogues in harmonic analysis: spherical averages
zbMATH Open document ID: 1036.42018
DOI: 10.2307/3062154
author: Ákos Magyar
author: Stephen Wainger
author: Elias M. Stein
published in: Annals of Mathematics. Second Series
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0409365
review text: This work is a continuation in a series of papers by the last two authors [cf., \textit{E. M. Stein} and \textit{S. Wainger}, Am. J. Math. 121, 1291--1336 (1999; Zbl 0945.42009); J. Anal. Math. 80, 335--355 (2000; Zbl 0972.42010)] pertaining to the study of discrete analogues of properties of singular integrals and maximal functions in harmonic analysis that depend, ultimately, on the geometry of the Euclidean space. The current paper considers a discrete analogue of the spherical average maximal operator \[ \sup_{\lambda>0} | f\ast d\sigma_\lambda| . \] This operator is bounded on \(L^p (\mathbb{R}^d)\) whenever \(p>d/(d-1)\) while \(d\geq 2\). Here \(d\sigma_\lambda\) denotes the normalized surface measure on the sphere of radius \(\lambda\). The discrete analogue of this operator takes the form \[ \sup_{\lambda>0}\frac{1}{N(\lambda)}\sum_{| m| =\lambda} f(n-m) \] where \(m,n\) take values in \(\mathbb{Z}^d\). This operator is bounded on a range of \(L^p\) spaces but, in contrast with the Euclidean case, it is necessary (and sufficient) that \(p>d/(d-2)\) while \(d\geq 5\) in order that this operator is bounded on \(\ell^p(\mathbb{Z}^d)\). As in previous results of the authors in this series of papers, considerations of Euclidean geometry like curvature need to be replaced by number theoretic concerns like class numbers. This is the principal reason for any differences in mapping properties. Analogues of ideas in sampling theory dating to Plancherel and Pólya also play a curious role here.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1791996
zbMATH Keywords: spherical averages
zbMATH Keywords: maximal functions
zbMATH Keywords: sampling
reviewed by: Joseph Lakey
MaRDI profile type: Publication
OpenAlex ID: W2003171351
arXiv ID: math/0409365
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description: scientific article