Partial Radon Transform and Hamburger moment completion in $\mathbb{R}^2$

The Radon transform is one of the most powerful tools for reconstructing data from a series of projections. Reconstruction of Radon transform with missing data can be closely related to reconstruction of a function from moment sequences with missing terms. A new range theorem is established for the Radon transform $Rf$ on $L^1$ based on the Hamburger moment problem in two variables, and the sparse moment problem is converted into the Radon transform with missing data and vice versa. A modified Radon transform for missing data is constructed and an inversion formula is established.