The support theorem for the Gauss-Radon transform
type: Item
instance of: scholarly article
zbMATH Open document ID: 1270.44002
DOI: 10.1142/S0219025712500063
author: Jeremy J. Becnel
published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics
publication date: 2012-08-30T00:00:00Z
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 60H40
zbMATH DE Number: 6074040
zbMATH Keywords: Radon transform
zbMATH Keywords: support theorem
zbMATH Keywords: Gaussian measure
zbMATH Keywords: infinite-dimensional distribution theory
zbMATH Keywords: white noise analysis
zbMATH Keywords: Gauss-Radon transform
zbMATH Keywords: infinite dimensions
zbMATH Keywords: Helgason support theorem
MaRDI profile type: Publication
cites work: Equivalence of topologies and Borel fields for countably-Hilbert spaces
cites work: Finite dimensional Hida distributions
cites work: White noise calculus and Fock space
full work available at URL: s0219025712500063
OpenAlex ID: W2026669265
title: The support theorem for the Gauss-Radon transform
review text: Recall the Radon transform which acts on functions \(f\) of the finite-dimensional space \(\mathbb{R}^n\) by NEWLINE\[NEWLINE R_f(P) = \int_P f(x)dx,NEWLINE\]NEWLINE where \(P\) is a hyperplane in \(\mathbb{R}^n\), and \(dx\) indicates integration with respect to Lebesgue measure on \(P\). The function \(R_f\) then acts on the set of all hyperplanes in \(\mathbb{R}^n\). The Radon transform does not immediately generalize to the setting of infinite dimensional spaces, since there is not a useful notion of Lebesgue measure in infinite dimensions. However, it is possible to define a Radon transform relative to Gaussian measures, for which there is a well-developed theorem in infinite dimensions. A Gauss-Radon transform may be defined by NEWLINE\[NEWLINE G_f(P) = \int f d\mu_P,NEWLINE\]NEWLINE where \(\mu_P\) is the Gaussian measure on a hyperplane \(P\). This is the setting in which an appropriate theory may be developed in infinite dimensions.NEWLINENEWLINEA key result in the finite dimensional setting is the Helgason support theorem for the Radon transform. This states that if \(f\) is a rapidly decreasing continuous function for which \(R_f(P)\) vanishes on every hyperplane \(P\) disjoint from some compact convex set \(K\), then \(f(x)=0\) for \(x \not\in K\). Rephrased for the Gauss-Radon transform, this result states that if \(f\) is an exponentially bounded continuous function for which \(G_f(P)\) vanishes on every hyperplane \(P\) disjoint from some compact convex set \(K\), then \(f(x)=0\) for \(x \not\in K\). The present paper proves the infinite-dimensional version of this support theorem for the Gauss-Radon transform in the setting of white noise analysis. This builds on previous work of the author and \textit{A. N. Sengupta} [Hackensack, NJ: World Scientific. QP-PQ: Quantum Probability and White Noise Analysis 22, 24--41 (2008; Zbl 1152.60052)].
reviewed by: Lillian Beatrix Pierce
description: scientific article; zbMATH DE number 6074040