Fourier transform of distributions Which distributions are their own Fourier transform besides the normal distribution and the generalized arcsine distribution?
 A: Suppose the Fourier transform of $x(t)$ is $X(f)$ where 
$$X(f) = \int_{-\infty}^{\infty} x(t) \exp(-i2\pi f t) \mathrm dt$$
where $i = \sqrt{-1}$.  The inverse transform is
$$x(t) = \int_{-\infty}^{\infty} X(f) \exp(i2\pi f t) \mathrm df$$
Some properties of the Fourier transform are as follows:


*

*The Fourier transform of $X(t)$ is $x(-f)$

*If $x(t)$ is a real-valued even function of $t$, then $X(f)$
is a real-valued even function of $f$.
Thus, if $x(t)$ is a real-valued even function of $t$, then
the Fourier transform of the real-valued even function 
$X(t)$ is $x(f)$
Now suppose that $x(t)$ is an even probability density function
(so that $x(t) \geq 0$ for all $t$) with the additional property 
that $x(0) = 1$. Suppose also that its Fourier transform $X(f)$ 
has the property that $X(f) \geq 0$ for all $f$.  Then, since
$$x(0) = 1 = \int_{-\infty}^{\infty} X(f) \mathrm df$$
$X(f)$ is a even non-negative real-valued function of $f$ with
area $1$, that is, $X(f)$ is also a probability density function
with the property that $X(0) = 1$. One example of such a pair of
functions is the normal distribution cited by OP Neil G
$$x_1(t) = \exp(-\pi t^2), ~~ X_1(f) = \exp(-\pi f^2)$$
and another example is
$$x_2(t) = (1 - |t|)\mathbf 1_{[-1,1]}, ~~ 
X_2(f) = \operatorname{sinc}^2(f) 
= \begin{cases}\displaystyle \left(\frac{\sin(\pi f)}{\pi f}\right)^2,&f\neq 0,\\
1,&f=0.\end{cases}$$
Now note that  $\frac{1}{2} x_2(t) + \frac{1}{2}X_2(t)$
is a mixture density whose Fourier transform is
$\frac{1}{2} X_2(f) + \frac{1}{2}x_2(f)$ which is 
the same mixture density.
Thus, if $x(t)$ is a density function whose
Fourier transform $X(f)$ is a density function, then
the mixture density function 
$\frac{1}{2} x(t) + \frac{1}{2}X(t)$ is its own Fourier transform.
Finally, given two densities that are their own
Fourier transforms, e.g. $x_1(t)$ and 
$\frac{1}{2} x_2(t) + \frac{1}{2}X_2(t)$,
any mixture density
$$\alpha x_1(t) + (1-\alpha)\left[\frac{1}{2} x_2(t) + \frac{1}{2}X_2(t)\right]$$
where $\alpha \in [0,1]$ is a density function that is its
own Fourier transform.
