On the Range of the Attenuated Radon Transform in Strictly Convex Sets

TL;DR

This paper establishes new necessary and sufficient conditions for functions to be in the range of the attenuated Radon transform within strictly convex sets, using an explicit Hilbert transform related to boundary traces of A-analytic functions.

Contribution

It introduces a novel characterization of the Radon transform's range in convex sets via an explicit Hilbert transform and boundary analysis of A-analytic functions.

Findings

Derived new range conditions for the attenuated Radon transform.

Connected boundary traces of A-analytic functions to the transform's range.

Provided a mathematical framework for reconstructing functions from Radon data.

Abstract

We present new necessary and sufficient conditions for a function on $\partial\Omega\times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set $\bar\Omega\subset\mathbb{R}^2$. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim.