# 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?