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Recovering functions from the spherical mean transform with data on an ellipse using eigenfunction expansion in elliptical coordinates

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Abstract

The aim of this paper is to introduce a new inversion procedure for recovering functions, defined on \(\mathbb R^{2}\), 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 Norton (J Acoust Soc Am 67:1266–1273, 1980) 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 [24] 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 Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the eigenfunction expansion of the Bessel function in elliptical coordinates.

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  1. Weizmann Institute of Science, Rehovot, Israel

    Yehonatan Salman

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  1. Yehonatan Salman
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Salman, Y. Recovering functions from the spherical mean transform with data on an ellipse using eigenfunction expansion in elliptical coordinates. Anal.Math.Phys. 9, 209–219 (2019). https://doi.org/10.1007/s13324-017-0192-6

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  • DOI: https://doi.org/10.1007/s13324-017-0192-6

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