infinite-dimensional distribution theory
The support theorem for the Gauss-Radon transform
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)].
scientific article; zbMATH DE number 6074040