I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception of one (which I've linked to in this post).

The integral I would like to investigate is the following:

$$\displaystyle \int_{\mathbb{R}^2} \frac{J_{1}(\rho \|x\|)J_{1}(\rho \|b-x\|)}{\|x\| \|b-x\|} \ \mathrm{d}x,$$

where $J_{\nu}$ denotes the Bessel function of the first kind, $b \in \Gamma \backslash \{0\}$ for any rational lattice $\Gamma \subset \mathbb{R}^2$ of full rank, $\rho > 0$ is large and independent of $b$ and $x$, and $\|\cdot\|$ denotes the standard Euclidean norm. I'll try to explain from where this integral arises, and why I care about it.

This integral arises when considering a problem on the distribution of lattice points inside a ball in $\mathbb{R}^d$ with radius $\rho$, which can be thought of as a generalisation of some aspects of the Gauss circle problem. For the full details, I recommend this paper (particularly pages 10-11 and 15-16). The paper considers $\sigma_p$ and proves some asymptotic bounds for $p = 1$ and $p = 2$ which depend on the dimension $d$ and the radius $\rho$. I'm attempting to generalise that work to $p = 4$, and with some extra work, any $p \in \mathbb{N}.$ The derivation of this integral begins here, and uses the estimate at the bottom of this post to arrive at the aforementioned integral. With the help of this answer, I was able to prove that the integral

$$\displaystyle \int_{\mathbb{R^d}} \|x\|^{-d/2}\|b-x\|^{-d/2}J_{d/2}(\rho \|x\|)J_{d/2}(\rho \|b-x\|) \ \mathrm{d}x,$$

converges absolutely in all dimensions $d \geqslant 3,$ after using the asymptotic bound for the Bessel function, $|J_{\nu}(z)| \leqslant C|z|^{-1/2},$ for $z \rightarrow \infty.$ However, as the answerer explains, the proof he provides does not work for the case $d = 2$. The integral at the top of this post is therefore the remaining case left to verify. I would like to know if it converges, and how in particular it can be estimated.

I don't have much experience in dealing with integrals involving Bessel functions. I would appreciate any help anyone can offer on how to deal with this integral.

  • 1
    $\begingroup$ Using your asymptotic bound, you can get $J_1(\rho\|b-x\|)/\|b-x\| \le C_b r^{-3/2}$, where $r = \|x\|$ and the constant is now allowed to depend on $b$. Plugging this bound into the integral in polar coordinates shows that it is dominated by $C_b \int^\infty dr/r^2$, which seems absolutely convergent to me. Am I missing anything? $\endgroup$ – Igor Khavkine Nov 5 '16 at 0:04
  • $\begingroup$ Forgive my ignorance, but I'm not sure I understand your answer. Firstly, could you explain how you can use the asymptotic bound to achieve that? Secondly, if you plug that bound into the integral with polar co-ordinates, doesn't $r$ range from $0$ to $\infty$ -- in which case, the integral diverges? $\endgroup$ – user363087 Nov 5 '16 at 13:48
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    $\begingroup$ First of all, the integrand is bounded and even continuous, so the only divergence can come from $r\gg 1$. So, if convergence of the integral is the only information you care about, there is no reason to bother with integrating the asymptotic estimates near the origin. Second, the only information that I used except the asymptotic inequality you already provided is that $1/\|x-b\| \le C_b 1/\|x\|$ for some constant $C_b$ and sufficiently large $\|x\|$. Hence, the integrand is a product, where each factor can be estimated in the same way. I think the rest follows easily. $\endgroup$ – Igor Khavkine Nov 5 '16 at 17:23

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