# Problem of correctly defining Hankel transforms

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?