On the injectivity of the circular Radon transform, centered on a curve, on $C(R^2)$

It is known that a continuous, compactly supported within a body $D$ (bounded and convex in $R^d$) function can be uniquely recovered from its spherical means over spheres centered on the boundary of $D$. In other words the boundary of $D$ is a set of injectivity of the spherical Radon transform on the space of compactly supported continuous functions.

An additional condition for the injectivity of the circular Radon transform on the space of all continuous functions on $R^2$ is found. This problem is equivalent to the injectivity of a weighted circular Radon transform. Also, a new iterative inversion algorithm for the weighted circular Radon transform is presented. This formula allows the reconstruction of an unknown function (not necessarily with compact support) using the local data of the circular integrals of the function over circles centered on a smooth curve. Such inversions are of theoretical importance in many areas of mathematics and are required in the mathematical models of thermo- and photo-acoustic tomography, radar imaging, and others.