Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions

A necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane. This problem is shown to be equivalent to inversion for the Radon transform on circles centered on restricted subsets of the plane. The proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular Radon transform. A characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems, and questions of Pinkus and Ehrenpreis are answered.