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
$$\hat{f}(\xi)=\int_{|x|\leq 1}e^{\dot{\imath}x_{1}|\xi|}e^{-\frac{1}{1-|x|^2}}dx=\int_{|x|\leq 1}e^{\dot{\imath}x_{1}|\xi|}e^{-\frac{1}{1-|x|^2}}dx\\= \int_{-1}^{1}e^{\dot{\imath}x_{1}|\xi|}\int_{|x^\prime|\leq \sqrt{1-x_1^2}}e^{-\frac{1}{1-x_1^2-|x^\prime|^2}}dx^\prime dx_1\\= \int_{-1}^{1}e^{\dot{\imath}x_{1}|\xi|}\int_{\mathbb{S}^{n-2}}\int_{0}^{\sqrt{1-x_1^2}}e^{-\frac{1}{1-x_1^2-\rho^2}}\rho^{n-2}d\rho d\omega_{n-2} dx_1\\ =|\mathbb{S}^{n-2}|\int_{-1}^{1}e^{\dot{\imath}x_{1}|\xi|}\int_{0}^{\sqrt{1-x_1^2}}e^{-\frac{1}{1-x_1^2-\rho^2}}\rho^{n-2}d\rho dx_1$$
And I am stuck here!