A support theorem for a Gaussian Radon transform in infinite dimensions (Q2880682)

The classical Radon transform associates to each function \(f\) of \(\mathbb{R}^n\) a function \(Rf\) acting on hyperplanes \(P\) in \(\mathbb{R}^n\) according to the definition NEWLINE\[NEWLINE (Rf)(P) = \int_P f(x) dx.NEWLINE\]NEWLINE Here the integration over the hyperplane \(P\) is with respect to Lebesgue measure on \(P\). A key aspect of the theory of the classical Radon transform is the Helgason support theorem: if \(f\) is a rapidly decreasing continuous function and \(Rf(P)\) vanishes for every hyperplane \(P\) lying outside a compact convex set \(K\), then \(f\) vanishes off \(K\). The goal of this paper is to prove an analogue of this support theorem for a version of the Radon transform acting on functions of infinite dimensional spaces.NEWLINENEWLINEThe first step in this extension is to define an appropriate Radon transform in the infinite dimensional setting. This is nontrivial; the authors consider the setting of a separable Hilbert space \(H_0\), in which case they define the Gauss-Radon transform NEWLINE\[NEWLINE(Gf)(P) = \int f d\mu_P,NEWLINE\]NEWLINE where \(\mu_P\) is Gaussian measure on the hyperplane \(P\) in the Hilbert space \(H_0\). (Note that in this setting, the term hyperplane refers to a translate of a closed linear subspace of codimension one.) The authors do a very careful job of defining the Gauss-Radon transform from first principles, including a detailed treatment of the measure \(\mu_P\) and the space upon which the function \(f\) is defined.NEWLINENEWLINEThe main theorem proved is as follows: Let \(H_0\) be a separable, infinite-dimensional real Hilbert space, which is a Hilbert-Schmidt completion of a Hilbert space \(H_1 \subset H_0\), and let \(H_{-1}\) be the dual to \(H_1\). Suppose that \(f: H_{-1} \rightarrow \mathbb{R}\) is a bounded function on the Hilbert space \(H_{-1}\) that is uniformly continuous in the strong (Hilbert) topology on \(H_{-1}\). Let \(K_0\) be a closed, bounded, convex subset of \(H_0\). If the Gauss-Radon transform of \(f\) vanishes on hyperplanes which do not intersect \(K_0\), then \(f\) vanishes on the complement of \(K_0\) in \(H_0\).NEWLINENEWLINEThis is proved via certain geometric observations, a ``disintegration process'' (i.e. a reduction to finite-dimensional subspaces) and a limiting argument. The authors note that the results of the paper extend to the setting of nuclear spaces. This is a very well-written paper and gives a good introduction to analysis on infinite dimensional spaces.