To simplify let us assume that $U \subset \mathbb{R}^n$ is an open set. Let $P$ be a (scalar) psuedodifferential operator of order $d$ and $p$ be its symbol. Therefore it may be expressed as follows: $Pf(x)=\int e^{ix \xi}p(x,\xi)\hat{f}(\xi)d\xi$ where $f \in C^{\infty}_c(U)$ is a smooth function with compact support. The fact that $P$ is of order $d$ imposes some growth condition on the function $p$.
Why the symbol may be reconstructed as $p(x,\xi)=e^{-ix \xi}P(e^{ix \xi})$.
This should be simple but I have a problem in understanding this formula since most likely it should interpreted in the distributional sense. I would be grateful for any help.
First, note that \begin{align} \widehat{e^{ix\eta}}(\xi) = \int dx\ e^{-ix(\xi-\eta)} = \delta(\xi-\eta) \end{align} then it follows \begin{align} P(e^{ix\eta})(x, \eta) =&\ \int d\xi\ e^{ix \xi} p(x, \xi)\ \widehat{e^{iy\eta}}(\xi)\\ =&\ \int d\xi\ e^{ix\xi}p(x, \xi) \delta(\xi-\eta)= e^{ix\eta}p(x, \eta). \end{align} Finally, we see that \begin{align} e^{-ix\eta}P(e^{ix\eta}) = p(x, \eta). \end{align}