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

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