Hankel transform with Bessel functions of the second kind

Question

The Hankel transform is defined for Bessel functions of the first kind (see e.g. http://en.wikipedia.org/wiki/Hankel_transform)

I would like to know if it is possible to define a Hankel transform with Hankel functions, or alternatively with Bessel functions of the second kind. It seems like a natural extension of the ordinary Hankel transform, but I have not been able to find any good references.

I know these functions are singular at the origin, but because the Hankel function is in some sense a natural construction, it seems like a reasonable thing to consider at least formally.

If this is possible, I would like to know in what cases it is useful, and if there are certain restrictions on the associated function space.

Thank you.

Answer 1

Some of the Bessel function class transforms. A general Google, or Google Scholar, search will yield some results linked to publications.

Hankel transform \begin{align} f(y) = \int_{0}^{\infty} f(x) \, J_{\nu}(xy) \, \sqrt{xy} \, dx \end{align}

Y-transform \begin{align} f(y) = \int_{0}^{\infty} f(x) \, Y_{\nu}(xy) \, \sqrt{xy} \, dx \end{align}

K-transform \begin{align} f(y) = \int_{0}^{\infty} f(x) \, K_{\nu}(xy) \, \sqrt{xy} \, dx \end{align}

Kontorovich-Lebedev transform \begin{align} f(y) = \int_{0}^{\infty} f(x) \, K_{i x}(y) \, dx \end{align}

H-transform \begin{align} \int_{0}^{\infty} f(x) \, {\bf{H}}_{\nu}(xy) \, \sqrt{xy} \, dx \end{align}