Let $f \in S(\mathbb{R}^2)$ and $g(x)= f(x,0)$. What is the relationship between $\widehat{g}$ and $\widehat{f}$?
It's clear that $g \in S(\mathbb{R})$ so both $f$ and $g$ have a Fourier transform. I expect something like $\widehat g(\xi)= \widehat f(\xi,0)$ but starting from the integral definitions gets me nowhere... Any ideas on how to proceed? Thanks!