The Paley–Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk $\mathbb{D}\subset\mathbb{R}^2$ from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk $\mathbb{D}'\subset\mathbb{D}$ under the additional condition that the curves that do not meet $\mathbb{D}'$ coincide with the corresponding straight lines.