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

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