The Spherical Mean Transform with Data on a Parabola in the Plane

Abstract: In this paper we deal with the problem of recovering functions from their spherical mean transform $\mathcal{R}$, which integrates functions on circles in the plane, in case where the centers of the circles of integration are located on a parabola $\mathcal{P}$ while their radii can be chosen arbitrarily. Using our data, on the values of $\mathcal{R}$ on $\mathcal{P}$, we show how to extract its values in the exterior of $\mathcal{P}$ in case where the functions in question have compact support inside $\mathcal{P}$. Hence, one can use known inversion formulas for $\mathcal{R}$ in the exterior of $\mathcal{P}$ in order to obtain a reconstruction formula.