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.
Categories: math.AP