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I have found the definition of the $v$-th order Hankel transform in the book of A.D. Poularikas "Transforms and applications" in Chapter $9$:

$$H_{v}(f(s))=\int_{0}^{+ \infty} rf(r) J_{v}(sr) dr$$

where $J_{v}$ is the $v$-th order Bessel function of the first kind.

My questions are:

  • If $f \in L^{2}(R^{+})$, is this transform well-defined? If it is the case $H_{v}(f)$ belongs to which set?.
  • How do we find the convergence of $J_{v}$? They put directly on other papers approximations of the Bessel function $J_{v}$. But what are the calculations to resolve those?
  • Last but not least what is $v$? Is it an integer or a real number?
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