 Hankel transform

The order-$\nu$ Hankel transform replaces the exponential kernel of a Fourier transform by a Bessel function:Under suitable integrability and inversion hypotheses, $f(r)={\int }_0^{\infty }({\mathcal{H}}_{\nu }f)(k)J_{\nu }(kr)kdk$. The measures $rdr$ and $kdk$ belong to this convention and cannot be discarded.

For a radial function $F(x)=f(|x|)$ on the plane, angular integration gives $\hat{F}(\xi )=2\pi ({\mathcal{H}}_{0}f)(|\xi |)$ when the Fourier transform kernel is $e^{-ix\cdot \xi }$. Thus rotational symmetry reduces a two-dimensional transform to a one-dimensional radial integral.