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Consider the following function in $\mathbb{R}^3$

$$ \begin{cases} 1 & |x|<1\\ 0 & \text{else} \end{cases}$$

How can we calculate its Fourier transform (again, in $\mathbb{R}^3$).

I tried to use spherical coordinates in order to calculate $$ \intop_{\left\{ x\in\mathbb{R}^{3}:|x|<1\right\} }e^{-i \langle \omega,x \rangle}\text{ }\,dx $$

But it seems impossible to finish the calculations.

I would like to see the right way to calculate this integral (because Im pretty sure using spherical coordinates is not the way although it seems the natural choice for a radial function in $\mathbb{R}^3$).

EDIT:

Followed up question:

As I understand from the post in the comments, there is no elementary solution for general $ n$. But, can we find an elementary solution (which does not involve Bessel's functions?)

FreeZe
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  • In $\mathbb R^1$, this would be some version of ${\sin x\over x}$, but in higher dimensions it is not as elementary... requiring some Bessel function (which does satisfy the "obvious" differential equation... which makes it not-so-not-elementary). – paul garrett Jun 18 '21 at 22:25
  • https://math.stackexchange.com/questions/489391/fourier-transform-of-the-indicator-of-the-unit-ball – N. S. Jun 18 '21 at 22:38
  • @paulgarrett For the particular case where $n=3 $, there exists an elementary solution? (not involving Bessel functions)? – FreeZe Jun 19 '21 at 06:21

0 Answers0