A symmetric integral identity for Bessel functions with applications to integral geometry

Abstract

In the article of Kunyansky (Inverse Probl 23(1):373–383, 2007) 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 ${\mathbb {R}}^{n}$. 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 ${\mathbb {R}}^{2}$. For this, we will use the recent results obtained by Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the expansions into eigenfunctions of Bessel functions of the first and second kind in elliptical coordinates.

Appendix

Lemma 4.1

Let $(\xi _{0}, \eta _{0})$ and $(\xi _{1}, \eta _{1})$ be two points in ${\mathbb {R}}^{2}$ and let $\xi _{2}$ be a positive real number such that $|\xi _{0}| < \xi _{2}$. Then, we have the following identity

$$\begin{aligned} u_{k}(\xi _{0}, \eta _{0}, \xi _{1}, \eta _{1})= & {} \frac{1}{4}\int _{-\pi }^{\pi }\left( v_{k}(\xi _{2}, \eta , \xi _{0}, \eta _{0})\partial _{1}u_{k}(\xi _{2}, \eta , \xi _{1}, \eta _{1})\right. \\&-\,\left. u_{k}(\xi _{2}, \eta , \xi _{1}, \eta _{1})\partial _{1}v_{k}(\xi _{2}, \eta , \xi _{0}, \eta _{0})\right) d\eta . \end{aligned}$$

Proof

The radial Bessel function of the second kind $Y_{0}(k|\cdot |)$ is a fundamental solution for the Helmholtz operator $\triangle _{x} + k^{2}$ in ${\mathbb {R}}^{2}$. That is, if

$$\begin{aligned} G(x) = \int _{{\mathbb {R}}^{2}}Y_{0}(k|x - y|)g(y)dy, \end{aligned}$$

(4.1)

then

$$\begin{aligned} \left( \triangle _{x} + k^{2}\right) G(x) = -4g(x) \end{aligned}$$

where g is any continuous function in ${\mathbb {R}}^{2}$ with compact support (see [11, page 3]). In elliptical coordinates the Helmholtz operator has the form

$$\begin{aligned} \triangle _{x} + k^{2} = \frac{1}{\cosh \xi ^{2} - \cos ^{2}\eta }\left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}}\right) + k^{2}. \end{aligned}$$

Hence, using the change of variables

$$\begin{aligned} y = x(\xi ', \eta '), \quad dy = \left( \cosh ^{2}\xi ' - \cos ^{2}\eta '\right) d\eta ' d\xi ' \end{aligned}$$

in Eq. (4.1) and then taking the Helmholtz operator we obtain that

$$\begin{aligned} -\, 4g\left( x(\xi , \eta )\right)= & {} \left( \frac{1}{\cosh \xi ^{2} - \cos ^{2}\eta }\left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}}\right) + k^{2}\right) G\left( x(\xi , \eta )\right) \\= & {} \left( \frac{1}{\cosh \xi ^{2} - \cos ^{2}\eta }\left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}}\right) + k^{2}\right) \\&\times \int _{0}^{\infty }\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')g(x(\xi ', \eta '))\left( \cosh ^{2}\xi ' - \cos ^{2}\eta '\right) d\eta ' d\xi ' \end{aligned}$$

where we used Eq. (1.13) relating the function $v_{k}$ with $Y_{0}$. This implies that

$$\begin{aligned}&= \left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}} + k^{2}\left( \cosh \xi ^{2} - \cos ^{2}\eta \right) \right) \\&\quad \times \int _{0}^{\infty }\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')g(x(\xi ', \eta '))\left( \cosh ^{2}\xi ' - \cos ^{2}\eta '\right) d\eta ' d\xi ' \\&= -\,4g\left( x(\xi , \eta )\right) \left( \cosh \xi ^{2} - \cos ^{2}\eta \right) . \end{aligned}$$

Thus, $v_{k}(\xi , \eta , \xi ', \eta ')$ is a fundamental solution, at the point $(\xi ', \eta ')$, for the differential operator

$$\begin{aligned} L_{\xi , \eta } = \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}} + k^{2}\left( \cosh \xi ^{2} - \cos ^{2}\eta \right) . \end{aligned}$$

Let $(\xi _{0}, \eta _{0}), (\xi _{1}, \eta _{1})$ be two arbitrary points in ${\mathbb {R}}^{2}$ such that $|\eta _{0}| < \pi$. Let us use Green’s Theorem for the functions $u_{k}\left( \partial _{2}v_{k}\right) - \left( \partial _{2}u_{k}\right) v_{k}$ and $\left( \partial _{1}u_{k}\right) v_{k} - u_{k}\left( \partial _{1}v_{k}\right)$, where

$$\begin{aligned} u_{k} = u_{k}\left( \cdot , \cdot , \xi _{1}, \eta _{1}\right) , \quad v_{k} = v_{k}\left( \cdot , \cdot , \xi _{0}, \eta _{0}\right) \end{aligned}$$

on the rectangle

$$\begin{aligned} R = \left\{ (\xi , \eta ):|\xi | < \xi _{2}, -\,\pi \le \eta \le \pi \right\} \end{aligned}$$

where we assume that $|\xi _{0}| < \xi _{2}$. Using the fact that $u_{k}$ is in the kernel of the operator $L_{\xi , \eta }$ (because of formula (1.12) and the fact that $J_{0}(k|\cdot |)$ is in the kernel of the Helmholtz operator $\triangle _{x} + k^{2}$) and fact that the rectangle R contains the singularity points $\pm \,(\xi _{0}, \eta _{0})$ of $v_{k}$ we have

$$\begin{aligned}&\oint _{\partial R}\left[ u_{k}\left( \partial _{2}v_{k}\right) - \left( \partial _{2}u_{k}\right) v_{k}\right] d\xi + \left[ \left( \partial _{1}u_{k}\right) v_{k} - u_{k}\left( \partial _{1}v_{k}\right) \right] d\eta \nonumber \\&\quad = \int _{R}\left[ \frac{\partial }{\partial \xi }\left( \frac{\partial u_{k}(\xi , \eta , \xi _{1}, \eta _{1})}{\partial \xi }v_{k}(\xi , \eta , \xi _{0}, \eta _{0}) - u_{k}(\xi , \eta , \xi _{1}, \eta _{1})\frac{\partial v_{k}(\xi , \eta , \xi _{0}, \eta _{0})}{\partial \xi }\right) \right. \nonumber \\&\qquad \left. -\frac{\partial }{\partial \eta }\left( u_{k}(\xi , \eta , \xi _{1}, \eta _{1})\frac{\partial v_{k}(\xi , \eta , \xi _{0}, \eta _{0})}{\partial \eta } - \frac{\partial u_{k}(\xi , \eta , \xi _{1}, \eta _{1})}{\partial \eta }v_{k}(\xi , \eta , \xi _{0}, \eta _{0})\right) \right] d\xi d\eta \nonumber \\&\quad = \int _{R}\left[ \left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}}\right) u_{k}(\xi , \eta , \xi _{1}, \eta _{1})v_{k}(\xi , \eta , \xi _{0}, \eta _{0})\right. \nonumber \\&\qquad \left. - \left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}}\right) v_{k}(\xi , \eta , \xi _{0}, \eta _{0})u_{k}(\xi , \eta , \xi _{1}, \eta _{1})\right] d\xi d\eta \nonumber \\&\quad = -\int _{R}u_{k}(\xi , \eta , \xi _{1}, \eta _{1})\left( \frac{\partial ^{2}}{\partial \xi ^{2}} + \frac{\partial ^{2}}{\partial \eta ^{2}} + k^{2}\left( \cosh \xi ^{2} - \cos ^{2}\eta \right) \right) v_{k}(\xi , \eta , \xi _{0}, \eta _{0})d\xi d\eta \nonumber \\&\quad = 4u_{k}(\xi _{0}, \eta _{0}, \xi _{1}, \eta _{1}) + 4u_{k}(-\,\xi _{0}, -\,\eta _{0}, \xi _{1}, \eta _{1}) = 8u_{k}(\xi _{0}, \eta _{0}, \xi _{1}, \eta _{1}). \end{aligned}$$

(4.2)

On the other hand, we can write more explicitly

$$\begin{aligned}&\oint _{\partial R}\left[ u_{k}\left( \partial _{2}v_{k}\right) - \left( \partial _{2}u_{k}\right) v_{k}\right] d\xi + \left[ \left( \partial _{1}u_{k}\right) v_{k} - u_{k}\left( \partial _{1}v_{k}\right) \right] d\eta \nonumber \\&\quad = \int _{-\xi _{2}}^{\xi _{2}}\left( \partial _{2} v_{k}(\xi , -\, \pi , \xi _{0}, \eta _{0})u_{k}(\xi , -\, \pi , \xi _{1}, \eta _{1})\right. \nonumber \\&\qquad \left. -\, \partial _{2}u_{k}(\xi , -\, \pi , \xi _{1}, \eta _{1})v_{k}(\xi , -\, \pi , \xi _{0}, \eta _{0})\right) d\xi \nonumber \\&\qquad + \, \int _{-\pi }^{\pi }\left( v_{k}(\xi _{2}, \eta , \xi _{0}, \eta _{0})\partial _{1} u_{k}(\xi _{2}, \eta , \xi _{1}, \eta _{1})\right. \nonumber \\&\qquad \left. -\, u_{k}(\xi _{2}, \eta , \xi _{1}, \eta _{1})\partial _{1} v_{k}(\xi _{2}, \eta , \xi _{0}, \eta _{0})\right) d\eta \nonumber \\&\qquad - \int _{-\xi _{2}}^{\xi _{2}}\left( \partial _{2} v_{k}(\xi , \pi , \xi _{0}, \eta _{0})u_{k}(\xi , \pi , \xi _{1}, \eta _{1})\right. \nonumber \\&\qquad \left. -\, \partial _{2}u_{k}(\xi , \pi , \xi _{1}, \eta _{1})v_{k}(\xi , \pi , \xi _{0}, \eta _{0})\right) d\xi \nonumber \\&\qquad - \int _{-\pi }^{\pi }\left( v_{k}(-\,\xi _{2}, \eta , \xi _{0}, \eta _{0})\partial _{1} u_{k}(-\,\xi _{2}, \eta , \xi _{1}, \eta _{1})\right. \nonumber \\&\qquad \left. -\, u_{k}(-\,\xi _{2}, \eta , \xi _{1}, \eta _{1})\partial _{1} v_{k}(-\,\xi _{2}, \eta , \xi _{0}, \eta _{0})\right) d\eta \nonumber \\&\quad = {\mathcal {I}}_{1} + {\mathcal {I}}_{2} + {\mathcal {I}}_{3} + {\mathcal {I}}_{4}. \end{aligned}$$

(4.3)

Observe that since $v_{k}$ and $u_{k}$ are $2\pi$ periodic then it follows that the integrals ${\mathcal {I}}_{1}$ and ${\mathcal {I}}_{3}$ cancel each other. Now, we claim that ${\mathcal {I}}_{2} = {\mathcal {I}}_{4}$. Indeed, since both $u_{k}(\cdot , \cdot , \xi _{1}, \eta _{1})$ and $v_{k}(\cdot , \cdot , \xi _{0}, \eta _{0})$ are even functions it follows that

$$\begin{aligned} v_{k}(-\,\xi , \eta , \xi _{0}, \eta _{0})= & {} v_{k}(\xi , -\,\eta , \xi _{0}, \eta _{0}),\\ u_{k}(-\,\xi , \eta , \xi _{0}, \eta _{0})= & {} u_{k}(\xi , -\,\eta , \xi _{0}, \eta _{0}). \end{aligned}$$

Hence, taking the derivative with respect to $\xi$ on both sides of the last two equations we obtain that

$$\begin{aligned} -\,\partial _{1}v_{k}(-\,\xi , \eta , \xi _{0}, \eta _{0})= & {} \partial _{1}v_{k}(\xi , -\,\eta , \xi _{0}, \eta _{0}),\\ -\,\partial _{1}u_{k}(-\,\xi , \eta , \xi _{0}, \eta _{0})= & {} \partial _{1}u_{k}(\xi , -\,\eta , \xi _{0}, \eta _{0}). \end{aligned}$$

Hence, we have

$$\begin{aligned} {\mathcal {I}}_{4}= & {} - \int _{-\pi }^{\pi }\left( v_{k}(-\,\xi _{2}, \eta , \xi _{0}, \eta _{0})\partial _{1}u_{k}(-\,\xi _{2}, \eta , \xi _{1}, \eta _{1}) \right. \\&\left. -\, u_{k}(-\,\xi _{2}, \eta , \xi _{1}, \eta _{1})\partial _{1}v_{k}(-\,\xi _{2}, \eta , \xi _{0}, \eta _{0})\right) d\eta \\= & {} \int _{-\pi }^{\pi }\left( v_{k}(\xi _{2}, -\,\eta , \xi _{0}, \eta _{0})\partial _{1}u_{k}(\xi _{2},\right. \\&\left. -\,\eta , \xi _{1}, \eta _{1}) - u_{k}(\xi _{2}, -\,\eta , \xi _{1}, \eta _{1})\partial _{1}v_{k}(\xi _{2}, -\,\eta , \xi _{0}, \eta _{0})\right) d\eta \end{aligned}$$

Thus, by making the change of variables $\eta \mapsto -\,\eta$ in the last integral we see that both integrals ${\mathcal {I}}_{2}$ and ${\mathcal {I}}_{4}$ coincide. Combining Eqs. (4.2) and (4.3) we obtain Lemma 4.1. $\square$

Lemma 4.2

Assume that $|\xi '| < \xi$. Then, the function $v_{k}$ has the following expansion

$$\begin{aligned} v_{k}(\xi ', \eta ', \xi , \eta )= & {} 2\pi i\mathrm {Mc}_{0}^{(1)}\left( \xi ', \frac{k}{2}\right) \mathrm {ce}_{0}\left( \eta ', \frac{k^{2}}{4}\right) \mathrm {Mc}_{0}^{(3)}\left( \xi , \frac{k}{2}\right) \mathrm {ce}_{0}\left( \eta , \frac{k^{2}}{4}\right) \\&+\, 2\pi i\sum _{n = 1}^{\infty }\left[ \mathrm {Mc}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \mathrm {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right) \mathrm {Mc}_{n}^{(3)}\left( \xi , \frac{k}{2}\right) \mathrm {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) \right. \\&\left. +\, \mathrm {Ms}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \mathrm {se}_{n}\left( \eta ', \frac{k^{2}}{4}\right) \mathrm {Ms}_{n}^{(3)}\left( \xi , \frac{k}{2}\right) \mathrm {se}_{n}\left( \eta , \frac{k^{2}}{4}\right) \right] \end{aligned}$$

into eigenfunctions in elliptical coordinates.

Proof

The proof of Lemma 4.2 is based on the proof of Theorem 5.2 in [7]. There, the authors derive the expansion into eigenfunctions in elliptical coordinates of the modified Bessel function of the second kind $K_{0}$. Here, we will use a slight modification of the proof introduced in [7] in order to adjust it to the Bessel function of the second kind $Y_{0}$.

Since $\xi '\ne \pm \,\xi$ then for every fixed $\xi , \xi '$ and $\eta '$ the function $f_{\xi , \xi ',\eta '}(z) = v_{k}(\xi , z, \xi ', \eta ')$ can be extended, in an open strip containing ${\mathbb {R}}$, into a $2\pi$ periodic analytic function. Thus, it has an expansion in the form of (1.9):

$$\begin{aligned} v_{k}(\xi , \eta , \xi ', \eta ')= & {} \alpha _{0}(\xi , \xi ', \eta ')\text {ce}_{0}\left( \eta , \frac{k^{2}}{4}\right) \\&+ \sum _{n = 1}^{\infty }\left( \alpha _{n}(\xi , \xi ', \eta ')\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) + \beta _{n}(\xi , \xi ', \eta ')\text {se}_{n}\left( \eta , \frac{k^{2}}{4}\right) \right) \end{aligned}$$

where

$$\begin{aligned} \alpha _{n}(\xi , \xi ', \eta ')= & {} \frac{1}{\pi }\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) d\eta ,\\ \beta _{n}(\xi , \xi ', \eta ')= & {} \frac{1}{\pi }\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')\text {se}_{n}\left( \eta , \frac{k^{2}}{4}\right) d\eta . \end{aligned}$$

Hence, our aim is to find the explicit form of the functions $\alpha _{n}$ and $\beta _{n}$.

Now, we are going to use again Lemma 4.1. This lemma was formulated for the function $u_{k}$, however it is clear from its proof that it is true when $u_{k}$ is replaced with any function $U(\xi , \eta )$ which is in the kernel of the operator $L_{\xi , \eta }$, even with respect to the variable $(\xi ,\eta )$ (i.e., $U(\xi , \eta ) = U(-\,\xi , -\,\eta )$) and which is $2\pi$ periodic with respect to the variable $\eta$. Hence, choosing $U(\xi , \eta ) = u(\xi )\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right)$ where u is an even function, which is a solution to (1.7), we have

$$\begin{aligned}&u(\xi ')\text {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right) \\&\quad = \frac{1}{4}\int _{-\pi }^{\pi }\left( u'(\xi )\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) v_{k}(\xi , \eta , \xi ', \eta ') \right. \\&\qquad \left. -\, \partial _{1} v_{k}(\xi , \eta , \xi ', \eta ')u(\xi )\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) \right) d\eta \end{aligned}$$

where the last identity is true in case where $|\xi '| < \xi$. Hence, if we define the following function

$$\begin{aligned} f(\xi ) = \frac{1}{4}\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) d\eta , \end{aligned}$$

(4.4)

then we have

$$\begin{aligned} u(\xi ')\text {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right) = u'(\xi )f(\xi ) - u(\xi )f'(\xi ). \end{aligned}$$

Let us choose $u(\xi ) = \text {Mc}_{n}^{(1)}\left( \xi , \frac{k}{2}\right)$, then we have

$$\begin{aligned} \text {Mc}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \text {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right)= & {} \partial _{\xi }\text {Mc}_{n}^{(1)}\left( \xi , \frac{k}{2}\right) f(\xi ) - \text {Mc}_{n}^{(1)}\left( \xi , \frac{k}{2}\right) f'(\xi )\nonumber \\= & {} W\left[ f, \text {Mc}_{n}^{(1)}\left( \cdot , \frac{k}{2}\right) \right] \end{aligned}$$

(4.5)

where W denotes the wronskian. Thus, the wronskian of f and $\text {Mc}_{n}^{(1)}$ is equal to a constant which does not depend on $\xi$. However, is was proved in [4, page 668, formula 28.20.1] that the wronskian of $\text {Mc}_{n}^{(1)}$ and $\text {Mc}_{n}^{(3)}$ is also equal to a constant. Hence, there exist constants $c_{1}$ and $c_{2}$ which do not depend on $\xi$ such that

$$\begin{aligned} f(\xi ) = c_{1}\text {Mc}_{n}^{(3)}\left( \xi ,\frac{k}{2}\right) + c_{2}\text {Mc}_{n}^{(1)}\left( \xi ,\frac{k}{2}\right) , \quad \xi > |\xi '|. \end{aligned}$$

However, we claim that $c_{2} = 0$. Indeed, taking the analytic extensions of the functions $f, \text {Mc}_{n}^{(3)}$ and $\text {Mc}_{n}^{(1)}$ in a narrow strip which contains the ray $[\xi ',\infty )$ we see, from Eqs. (1.10) and (4.4), that both f and $\text {Mc}_{n}^{(3)}$ have the same asymptotic expansion at infinity (observe that $f(z)\approx Y_{0}(k\cosh z)$) while $\text {Mc}_{n}^{(1)}$ behaves like [see Eq. (1.11)] $e^{\left| \mathfrak {I}\left( k\cosh z\right) \right| }O\left( \left( \cosh z\right) ^{-\frac{3}{2}}\right)$. This implies that $c_{2} = 0$ and hence

$$\begin{aligned} f(\xi ) = c_{1}(\xi ', \eta ')\text {Mc}_{n}^{(3)}\left( \xi ,\frac{k}{2}\right) , \quad \xi > |\xi '|. \end{aligned}$$

(4.6)

In order to find the constant $c_{1}$ we use the fact that the wronskian of $\text {Mc}_{n}^{(1)}$ and $\text {Mc}_{n}^{(3)}$ is equal to $-\,2i / \pi$ and Eq. (4.5) in order to obtain that

$$\begin{aligned} c_{1} = -\frac{\pi }{2i}\text {Mc}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \text {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right) . \end{aligned}$$

Hence, using Eqs. (4.4) and (4.6) we finally obtain that

$$\begin{aligned}&\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')\text {ce}_{n}\left( \eta , \frac{k^{2}}{4}\right) d\eta \\&\quad = 2\pi i\text {Mc}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \text {ce}_{n}\left( \eta ', \frac{k^{2}}{4}\right) \text {Mc}_{n}^{(3)}\left( \xi ,\frac{k}{2}\right) ,\quad |\xi '| < \xi . \end{aligned}$$

In the exact same way we can show that

$$\begin{aligned}&\int _{-\pi }^{\pi }v_{k}(\xi , \eta , \xi ', \eta ')\text {se}_{n}\left( \eta , \frac{k^{2}}{4}\right) d\eta \\&\quad = 2\pi i\text {Ms}_{n}^{(1)}\left( \xi ', \frac{k}{2}\right) \text {se}_{n}\left( \eta ', \frac{k^{2}}{4}\right) \text {Ms}_{n}^{(3)}\left( \xi ,\frac{k}{2}\right) ,\quad |\xi '| < \xi . \end{aligned}$$

This proves Lemma 4.2. $\square$

Lemma 4.3

Let f be a continuous function defined on ${\mathbb {R}}^{2}$ with compact support, then f has the following representation

$$\begin{aligned} f(x) = \frac{1}{2\pi }\int _{0}^{\infty }\int _{{\mathbb {R}}^{2}}f(y)J_{0}(k|x - y|)dykdk. \end{aligned}$$

Proof

From the inversion formula for the Fourier transform in ${\mathbb {R}}^{2}$ we obtain that

$$\begin{aligned} f(x) = \frac{1}{2\pi }\int _{{\mathbb {R}}^{2}}{\mathcal {F}}(f)(y)e^{ix\cdot y}dy \end{aligned}$$

where ${\mathcal {F}}(f)$ denotes the Fourier transform of f. Using the definition of the Fourier transform we have

$$\begin{aligned} f(x)= & {} \frac{1}{4\pi ^{2}}\int _{{\mathbb {R}}^{2}}\int _{{\mathbb {R}}^{2}}f(z)e^{-iz\cdot y}dze^{ix\cdot y}dy = \frac{1}{4\pi ^{2}}\int _{{\mathbb {R}}^{2}}\int _{{\mathbb {R}}^{2}}f(z)e^{i(x - z)\cdot y}dydz \\= & {} \left[ y = ke^{i\theta }, dy = kd\theta dk\right] = \frac{1}{4\pi ^{2}}\int _{{\mathbb {R}}^{2}}\int _{0}^{\infty }\int _{-\pi }^{\pi }f(z)e^{ik(x - z)\cdot e^{i\theta }}d\theta kdkdz \\= & {} \frac{1}{2\pi }\int _{0}^{\infty }\int _{{\mathbb {R}}^{2}}f(z)J(k|x - z|)dzkdk \end{aligned}$$

where in the last passage we used the following formula

$$\begin{aligned} J_{0}(t) = \frac{1}{2\pi }\int _{-\pi }^{\pi }e^{it\cos \theta }d\theta \end{aligned}$$

for the Bessel function of the first kind. This proves Lemma 4.3. $\square$