About FBI transform

Question

I read a statment on a book saying that the FBI transform $$\mathcal{F}_u(x,\xi)=\int_{\mathbb{R}^m} e^{i\xi.(x-y)-|\xi||x-y|^2}u(y)\,dy ,\; (x,\xi)\in \mathbb{R}^m \times \mathbb{R}^m$$ is nonlinear. I can not see how this is nonlinear.

Answer 1

The term ''nonlinear'' is used to indicate the non-linearity in the exponent,i.e, $|x-y|^2$