Recovering Functions from the Spherical Mean Transform with Data on an Ellipse Using Eigenfunction Expansion in Elliptical Coordinates

Abstract

The aim of this paper is to introduce a new inversion procedure for re- covering functions, defined on $\Bbb R^{2}$, from the spherical mean transform, which integrates functions on a prescribed family $\Lambda$ of circles, where $\Lambda$ consists of circles whose centers belong to a given ellipse E on the plane. The method presented here follows the same procedure which was used by S. J. Norton in [22] for recovering functions in case where $\Lambda$ consists of circles with centers on a circle. However, at some point we will have to modify the method in [22] by using expansion in elliptical coordinates, rather than spherical coordinates, in order to solve the more generalized elliptical case. We will rely on a recent result obtained by H.S. Cohl and H.Volkmer in [8] for the eigenfunction expansion of the Bessel function in elliptical coordinates.