On the range of the planar $X$-ray transform on the Fourier lattice of the torus

TL;DR

This paper characterizes the Fourier coefficients of functions on the torus that are in the range of the planar X-ray transform, linking classical and modern range descriptions.

Contribution

It provides necessary and sufficient conditions for Fourier coefficients to belong to the X-ray transform range, connecting Bukhgeim-Hilbert and classical characterizations.

Findings

Established range conditions based on Fourier coefficients

Connected Bukhgeim-Hilbert transform with classical range characterizations

Extended understanding of the X-ray transform on the torus

Abstract

We find necessary and sufficient conditions on the Fourier coefficients of a function $g$ on the torus to be in the range of the $X$-ray transform of functions with compact support in the plane, and establish the connection between the range characterization based on the Bukhgeim-Hilbert transform and the classical Gelfand-Graev, Helgason, and Ludwig characterization.