# Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function has a radial profile given by the Hankel transform of order $\nu=\frac{N}{2}-1$ of the input radial profile.

However, he does not include any citation, and does not prove the result (the chapter is mainly devoted to the case $N=2$). I believe it is wrong, because in the case $N=3$, we have $$\mathcal{F}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)r^2 j_0(kr)\,\mathrm{d}r\\=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)r\frac{\sin(kr)}{k}\,\mathrm{d}r$$ where $j_0$ is the zeroth order spherical Bessel function, whereas $$\mathcal{H}_{1/2}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)=\int_0^\infty f(r)rJ_{1/2}(kr)\,\mathrm{d}r=\sqrt{\frac{2}{\pi}}\int_0^\infty f(r)\sqrt{r}\frac{\sin(kr)}{\sqrt{k}}\,\mathrm{d}r$$ which are not the same.

Is this an error? If so, what is the correct $N$-dimensional analogue of the Fourier transform of radially-symmetric functions?

## 1 Answer

I actually also struggled a lot with this one ;)

The correct formula can be found for example in (Grafakos & Teschl, 2013)

$$\mathcal{F}\left(f(|\mathbf{r}|),\mathbf{r},\mathbf{k}\right)= \frac{2\pi}{k^{N/2-1}}\mathcal{H}_{N/2-1}(f(r)r^{N/2-1},r,2\pi k).$$

Grafakos, L., & Teschl, G. (2013). On Fourier transforms of radial functions and distributions. Journal of Fourier Analysis and Applications, 19(1), 167-179.

• If you could copy the formula from the arxiv paper into this answer (with citation), that would be very useful. Links can "rot" over time (i.e. webpages get deleted), which means that answers which rely on links for primary content can become outdated. Copying the relevant formula into this answer enables the answer to be useful past the lifetime of the linked page. – apnorton Jan 15 '15 at 21:47