A Symmetric Integral Identity for Bessel Functions with Applications to Integral Geometry

TL;DR

This paper extends a symmetric integral identity for Bessel functions to elliptical coordinates, enabling explicit inversion formulas for spherical mean transforms on ellipses, with applications in integral geometry.

Contribution

It introduces an analogous symmetric integral identity for Bessel functions in elliptical coordinates, expanding the applicability of inversion formulas in integral geometry.

Findings

Derived a new symmetric integral identity for Bessel functions on ellipses.

Provided explicit inversion formulas for spherical mean transforms on elliptical domains.

Connected recent eigenfunction expansion results to practical integral geometry applications.

Abstract

In the article [11] of L. Kunyansky a symmetric integral identity for Bessel functions of the first and second kind was proved in order to obtain an explicit inversion formula for the spherical mean transform where our data is given on the unit sphere in $\Bbb R^{n}$. The aim of this paper is to prove an analogous symmetric integral identity in case where our data for the spherical mean transform is given on an ellipse $E$ in $\Bbb R^{2}$. For this, we will use the recent results obtained by H.S. Cohl and H.Volkmer in [7] for the expansions into eigenfunctions of Bessel functions of the first and second kind in elliptical coordinates.