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I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. The $d$-dimensional Fourier transform is

\begin{align*} \widehat{f}(\xi) &= \int_{\mathbb{R}^d}f(x) e^{-2\pi i \xi \cdot x}\text{d} x\\ &= \int_0^\infty f(r) r^{d-1}\int_{S^{d-1}} e^{-2\pi i r \xi \cdot s}\text{d} s \text{d} r \end{align*} where $f(r)$ denotes the value on the radius-$r$ sphere. This is the integral of $f$ times the function $$ \psi_r(\xi) := r^{d-1}\int_{S^{d-1}} e^{-2\pi i r \xi \cdot s}\text{d} s$$ which is also radial so it's equivalent to a function $\psi_r : \mathbb{R}_{\geq 0} \to \mathbb{R}$.

My question is, do the functions $\psi_r$ have a name? Are they somehow less natural than the Bessel functions?

If I did the calculation correctly, the formula in terms of Bessel functions of the first kind $J_\nu(x)$ is $$ \psi_r(x) = \frac{r^{\frac{d}{2}}\Gamma(\frac d 2)J_{\frac{d-2}{2}}(2\pi rx)}{(\pi x)^{\frac{d-2}{2}}}$$

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    $\begingroup$ it's a Bessel function, why would it need a different name? $\endgroup$
    – Carlo Beenakker
    Commented Apr 9 at 14:56
  • $\begingroup$ It's less important whether it has a different name, but whether other works perform analysis using the $\psi_r$ perspective. $\endgroup$
    – Chris Jones
    Commented Apr 9 at 15:35
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    $\begingroup$ you might want to be more specific in what you are asking; what you have written down is the Hankel transform of order $d/2-1$ of the function $f(r)r^{d/2-1}$; what else is there to say? $\endgroup$
    – Carlo Beenakker
    Commented Apr 9 at 16:32

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The function does not have a standard term but is widely understood to be the radial solution of the Helmholtz equation (which is the time-periodic solutions of the wave equation) arrived at by what is known as the "method of spherical means". In Fritz "Plane Waves and Spherical Means" (Springer 1981) p77-78 he says your function "is a kind of "normalized" Bessel function". Compare your result to his equation (4.4) p.78. You can also find your equation in Courant & Hilbert "Methods of Mathematical Physics Vol II" (Wiley 1962) p.289 equation (39).

The Bessel function of integer and fractional order behave very differently and specifically the half fractions turn out to reduce to elementary functions (compare Courant & Hilbert ibid, top equation on page 290.).

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