## 2D wave front set: simple planar shapes (in progress)

Let $$h>0$$ be the semiclassical parameter. The FBI (Fourier-Bros-Iagolnitzer) transform of an $$n$$-dimensional tempered distribution $$u \in \mathscr{S}^\prime(\mathbb{R}^n)$$ is defined by $T_hu(x;\xi) := 2^{-n/2} (\pi h)^{-3n/4} \int_{\mathbb{R}^n} e^{-i(x-y)\cdot\xi/h-\lvert{x-y}\rvert^2/2h} u(y) dy,$ where $$(x,\xi) \in T^\ast\mathbb{R}^n\simeq\mathbb{R}^n\times\mathbb{R}^n$$. It is well-known that if $$u$$ is independent of $$h>0$$, then the wave front set of $$u$$ is characterized by its FBI transform. In fact it follows that $$(x_0,\xi_0) \in T^\ast\mathbb{R}^n\setminus0$$ does not belong to $$\operatorname{WF}(u)$$ if and only if $T_hu(x;\xi)=\mathcal{O}(h^\infty) \quad (h \downarrow0)$ uniformly near $$(x_0,\xi_0)$$. Note that a wave front set is conic in $$\xi$$ and independent of the size of $$\xi$$. We are trying to detect the wave front sets of 2D distributions using MATLAB and the FBI transform. For $$u(x,y) \in \mathscr{S}^\prime(\mathbb{R}^2)$$ we compute $T_hu(x,y;\lambda\cos\theta,\lambda\sin\theta) = 2^{-1}(\pi h)^{-3/2} \iint_{\mathbb{R}^2} e^{i\lambda(p\cos\theta+q\sin\theta)/h-(p^2+q^2)/2h} u(x+p,y+q) dpdq$ where $$\lambda$$ is the size of the frequency.

 Square (square.m) Right-angled isosceles triangle (triangle.m) Circle (singsuppcircle.m)