It's well-known that the Fourier transformation of a Gaussian distribution is still a Gaussian, \begin{align*} \int dx~ e^{-\frac{A}{2} x^2} e^{-ikx} = \int dx~ e^{-\frac{A}{2} (x+i \frac{k}{A})^2} e^{-\frac{k^2}{2A}} \propto e^{-\frac{k^2}{2A}}, \end{align*} where the width in $x$-space is inversely proportional to the width in $k$-space. In physics, this can be viewed as the uncertainty principle of the position-momentum duality.
I am curious whether there exists a distribution on discrete periodic space such that similar property holds. More concretely, for $x=0,1,2,...,n-1$ with $x=n$ being identified with $x=0$, whether there exist a periodic function $F_A(x)$ where $A\in \mathbb{R}$ is used to control the width of the distribution, such that \begin{align*} \sum\limits_{x=0}^{n-1} F_A(x) e^{-i (\frac{2\pi}{n}l )x} \propto F_{A'}(l) \end{align*} where $l\in 0,1,2,...,n-1$ is the discrete Fourier mode and $A'\in \mathbb{R}$ is controlling the dual width in the Fourier space, supposedly monotonically decreasing as $A$ increases.
Thanks!