On the range of the $X$-ray transform of symmetric tensors compactly supported in the plane

TL;DR

This paper characterizes the exact conditions on Fourier coefficients of functions on the torus that determine whether they are in the range of the X-ray transform of symmetric tensors with compact support in the plane.

Contribution

It provides necessary and sufficient Fourier coefficient conditions for the X-ray transform range of symmetric tensors in the plane, advancing understanding of tensor tomography.

Findings

Derived explicit Fourier coefficient conditions for the X-ray transform range.

Established a complete characterization of the transform's range for symmetric tensors.

Enhanced the theoretical foundation for tensor tomography in planar settings.

Abstract

We find the 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 a symmetric tensor of compact support in the plane.