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 $n \ge 3$?

Generalized functions. You will find and explicit description of a distribution, concentrated along the unit sphere, whose Fourier transform is $(\sin|\xi|)/|\xi|$ $\endgroup$ – Liviu Nicolaescu Nov 18 '18 at 17:06