The Radon transform in infinite-dimensional spaces
The classical Radon transform of a scalar function \(f\) on \(\mathbb R^n\) is defined on the space of affine subspaces of codimension 1, i.e., on the space of sets of the form \(L+x\), where \(L\) is a hyperplane in \(\mathbb R^n\), and acts by the formula \[ {\mathcal R} f(L+x)= \int_{L+x} f(u)du. \] The goal of this paper is to introduce the Radon transform for measures on infinite-dimensional spaces. Such a transform has been discussed before by A. Hertle, J. J. Becnel, and A. N. Sengupta, and V. Mihai. The authors of the paper under review propose a new definition that appears to have analogous properties of the classical Radon transform. To this end, the authors introduce the concept of a Radon measure on a locally convex space \(X\) and a system of conditional measures. They consider the restriction of a Radon measure to the \(\sigma\)-algebra \(\sigma (X^\ast)\) generated by the space \(X^\ast\) of continuous linear functionals on \(X.\) Let \(l\in X^\ast\), \(l\neq 0\), \(L=\text{Ker }l=l^{-1}(0)\) and pick \(v\in X\) with \(l(v)=1.\) The affine subspaces \(L+tv\) include all affine subspaces of the form \(L+x\), \(x\in X\). Under the assumption that there exist conditional measures \(\mu^{L+x}\) on the sets \(L+x\) satisfying certain conditions, the Radon transform of a bounded Borel function \(f\) on \(X\) is defined by the formula \[ {\mathcal R}^{\mu}f(L+x)=\int_{X}f(y)\mu^{L+x}\left( dy\right) . \] The authors discuss properties of this transform and compare it with other definitions given by the authors cited above. The paper is concluded with a uniqueness theorem for the Radon transform \({\mathcal R}^{\mu}f\).
scientific article; zbMATH DE number 6106423