Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$?


It is known that for $n = 2$, the function $\displaystyle f(x) = \frac{\chi\_{\{|x| < 1\}}}{\sqrt{1-|x|^2}}$ curiously has some constant multiple of $\dfrac{\sin |\xi|}{|\xi|}$ as Fourier transform. Or at least is there a way of finding the Fourier transform $\frac{\sin |\xi|}{|\xi|}$ in sense of tempered distributions for $n \ge 3$?



Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?
Here's a formula from Duistermaat and Kolk book *Distributions: Theory and Applications*, Chapter 17, Eq, (17.13). We denote by $\newcommand{\eF}{\mathscr{F}}$ $\eF$ the Fourier transform. Then $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\ve}{\varepsilon}$


$$
\eF^{-1}\left(\frac{e^{\ii t\Vert \xi\Vert}}{\Vert\xi\Vert}\right)=p\_n\lim\_{\ve\searrow 0} \Big(\; \Vert x\Vert^2-(t+\ve\ii)^2\;\Big)^{-\frac{n-1}{2}},
$$
where $p\_n$ is a certain universal constant,


$$p\_n=\frac{\Gamma\big(\frac{n+1}{2}\big)}{(n-1)\pi^{(n+1)/2}}. $$


