16
$\begingroup$

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.

$\endgroup$
  • $\begingroup$ Of course, it depends on $n$. $\endgroup$ – limanac Sep 10 '13 at 12:36
  • $\begingroup$ The unit ball for the Euclidian norm I guess. $\endgroup$ – Davide Giraudo Sep 10 '13 at 12:39
21
$\begingroup$

Let $\alpha_d = \dfrac{\pi^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$ the volume of the $d$-dimensional unit ball. Since the characteristic function of the unit ball is rotationally symmetric, so is its Fourier transform, hence let's compute it at the point $\xi = (0,\,\dotsc,\,0,\,\rho)$ with $\rho > 0$:

$$\begin{align} \hat{\chi}_B(\xi) &= \frac{1}{(2\pi)^{n/2}} \int_{\lVert x\rVert < 1} e^{-i x_n\rho}\, dx_1\, \dotsc\, dx_n\\ &= \frac{1}{(2\pi)^{n/2}} \int_{-1}^1 \left(1-x_n^2\right)^{(n-1)/2}\alpha_{n-1} e^{-i x_n\rho}\,dx_n\\ &= \frac{1}{2^{n/2}\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)} \int_0^\pi \sin^n \varphi e^{-i\rho\cos\varphi}\,d\varphi. \end{align}$$

Recalling that we have the Bessel functions

$$J_p(x) = \frac{(x/2)^p}{\sqrt{\pi}\Gamma\left(p + \frac12\right)}\int_0^\pi \sin^{2p} \varphi e^{-ix\cos\varphi}\,d\varphi,$$

we see that

$$\hat{\chi}_B(\xi) = \lVert \xi\rVert^{-n/2}J_{n/2}(\lVert\xi\rVert).$$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.