Fourier analysis – Wigner distribution


The Wigner distribution of $ u in L ^ 2 ( mathbb R) $ is defined as a function $ W (u) $ on $ mathbb R ^ 2 $ given by
$$
W (u) (x, xi) = int_ mathbb R u (x + frac z2) overline {u (x- frac z2)} e ^ {- 2π i z xi} dz.
$$

It's easy to see $ W (u) $ heard $ L ^ 2 ( mathbb R ^ 2) $ With
$
Vert W (u) Vert_ {L ^ 2 ( mathbb R ^ 2)} = Vert u Vert_ {L ^ 2 ( mathbb R)} ^ 2
$

since $ W (u) $ is the partial Fourier transform $ z $ from $ (x, z) mapsto u (x + frac z2) overline {u (x- frac z2)} $.

However, I believe that for some $ u in L ^ 2 ( mathbb R) $, the function $ W (u) $ is not one of them
$ L ^ 1 ( mathbb R ^ 2) $. Is there an "explicit" $ u $ like this?