Support theorems for the Radon transform and Cram\'er-Wold theorems

TL;DR

This paper extends the Cramér-Wold theorem to measures with infinite mass near the origin, using Radon transform injectivity results, distribution theory, and Fourier analysis, with sharp examples and applications to probability theory.

Contribution

It introduces new Cramér-Wold type theorems for measures with infinite mass near zero, based on Radon transform injectivity and distribution analysis.

Findings

Extensions of Cramér-Wold theorem to measures with infinite mass near zero

Injectivity results for Radon transform applied to measure characterization

Examples demonstrating sharpness of assumptions

Abstract

This article presents extensions of the Cram{\'e}r-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distribution theory and Fourier analysis we show that the presented injectivity results for the Radon transform lead to Cram{\'e}r-Wold type results for measures. One purpose of this article is to contribute to making known to probabilists interesting results for the Radon transform that have been developed essentially during the 1980ies and 1990ies.