What is the Fourier transform of the bump function $e^{-frac{1}{1-|x|^2}}$?

Mathematics Asked by Medo on January 5, 2022

Let
$$f(x):= left{ begin{array}{ll} e^{-frac{1}{1-|x|^2}}, & hbox{|x|<1;} \ 0, & hbox{|x|geq1.} end{array} right.$$

This is a generic bump function (a smooth positive bounded function with compact support). It is a Schwartz function, so it has a Schwartz Fourier transform.

My question is about calculating its Fourier transform $$hat{f}$$:

Since $$f$$ is radial (i.e. rotationally invariant), then so is $$hat{f}$$. So $$hat{f}(xi)=hat{f}(|xi|,0,…,0)$$ for all $$xi in mathbb{R}^{n}$$. Therefore, denoting $$x^prime=(x_2,x_3,…,x_n)$$, we have

$$hat{f}(xi)=int_{|x|leq 1}e^{dot{imath}x_{1}|xi|}e^{-frac{1}{1-|x|^2}}dx=int_{|x|leq 1}e^{dot{imath}x_{1}|xi|}e^{-frac{1}{1-|x|^2}}dx\= int_{-1}^{1}e^{dot{imath}x_{1}|xi|}int_{|x^prime|leq sqrt{1-x_1^2}}e^{-frac{1}{1-x_1^2-|x^prime|^2}}dx^prime dx_1\= int_{-1}^{1}e^{dot{imath}x_{1}|xi|}int_{mathbb{S}^{n-2}}int_{0}^{sqrt{1-x_1^2}}e^{-frac{1}{1-x_1^2-rho^2}}rho^{n-2}drho domega_{n-2} dx_1\ =|mathbb{S}^{n-2}|int_{-1}^{1}e^{dot{imath}x_{1}|xi|}int_{0}^{sqrt{1-x_1^2}}e^{-frac{1}{1-x_1^2-rho^2}}rho^{n-2}drho dx_1$$

And I am stuck here!

Related Questions

Which functions are spherical derivatives?

1  Asked on December 22, 2020 by giuseppe-negro

divergence of the cross product of two vectors proof

2  Asked on December 22, 2020 by mathematicing

What is the difference between radical ideal and the radical of an ideal?

1  Asked on December 22, 2020

How to understand the Galois *-action on a Dynkin diagram

1  Asked on December 22, 2020 by joshua-ruiter

Shape Transformation Via Medial Axis Transform

0  Asked on December 22, 2020 by fweth

Showing that $mathbb{C}[x,y]^{mu_n}$ and $mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic as rings

1  Asked on December 22, 2020

How do I rotate 18 people into pairs so that each person gets to talk to every other person without any overlap

0  Asked on December 22, 2020 by jimk

Is there a sequence of rational numbers $a_n$ such that $a_1 e^{-1} + a_2 e^{-2} + cdots = 1$?

1  Asked on December 22, 2020 by nilos

Regarding Lemma 21.9 of Jech

1  Asked on December 22, 2020 by hanul-jeon

Doppler effect mathematical modeling

0  Asked on December 21, 2020 by general-mo7

Showing that a group $G$ such that 3 does not divide $|G|$ is Abelian.

1  Asked on December 21, 2020 by user778657

Probability of getting a hand with at least 6 doubles in domino

0  Asked on December 21, 2020 by rolando-gonzlez

Prove that the composition map is continuous with respect to the metric topology on $operatorname{Iso}(M)$

1  Asked on December 21, 2020 by abhigyan-saha

Exactly Half of a 3D Transformation matrix

1  Asked on December 21, 2020 by sai-manoj-prakhya

Range of quadratic function using discriminant

3  Asked on December 21, 2020 by mutse

Why did we call a row operation “elementary”?

1  Asked on December 21, 2020 by neothehero

Let 4 linearly dependent vectors such that any 3 of them is independence

1  Asked on December 20, 2020 by roach87

A computation in the field of rational functions.

1  Asked on December 20, 2020 by jake-mirra

Why we cannot extend real numbers with dedekind cuts?

1  Asked on December 20, 2020 by thisguy

How do you find a point on a line bisecting an angle in three-dimensional space?

4  Asked on December 20, 2020 by user2561523