## For which $p(k)$ does $\int_0^{\infty} \int_0^\infty p(k) J_n(k r) \ k\ r\ dk \ dr$ converge

This is a follow up to a question on math.stackexchange

Consider the following integral,

$\int_0^{\infty} \int_0^\infty p(k) J_n(k r)\ k\ r\ dk\ dr$

where $k$ and $r$ are real, $n$ is a positive integer and $p(k) \in L^2(\mathbb{R})$. $p(k)$ is positive and decays to 0 at $k = 0$ and $k = \infty$ (a bandpass filter spectral envelope).

For which $p(k)$ does the integral converge? If it does converge, is it true that

$1/n \int_0^{\infty} \int_0^\infty p(k) J_n(k r)\ k\ r\ dk\ dr = 1/m \int_0^{\infty} \int_0^\infty p(k) J_m(k r)\ k\ r\ dk\ dr$

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