Does anyone know what is the Fourier transform of$$ f(x)=e^{-i/x} $$on the real line? I would like to compute it explicitly, or to establish some properties to have a good feeling of “how it looks like”.
I know that $\widehat f$ is real-valued by the symmetries of the Fourier transform, since $f(-x)=\overline{f(x)}$. It is natural to consider $h(x)=f(x)-1$. It is immediate that $h$ decays like $-i/x$ for large $x$, so $\widehat h$ belongs to $L^p(\mathbb R)$ for any $p\in [2,\infty)$ by Hausdorff-Young inequality. Then $\widehat f$ is recovered just by adding a Dirac delta.
Motivation.(not necessary to understand the problem)
For some reason, I was looking at the “anti-transport” equation$$ u_t-\partial_x^{-1}u=0. $$Even if it does not entirely make sense, one can consider the unitary group $e^{t\partial_x^{-1}}$, which is well-defined on $L^2$, that acts on the Fourier side as a multiplication by $e^{-it/\xi}$. The kernel of this PDE coincides with a rescaled version of the anti-Fourier transform of $e^{-i/\xi}$, this is why I considered the above problem. I was wondering what is the time-regularity of solutions of such PDE, even for smooth initial data: I find a bit funny that the operator $\partial_x^{-1}$ should in principle “smoothen things out” if well-defined, but it seems that the time-derivative of a solution of the PDE is not necessarily smooth for smooth data, unless one imposes some low-frequency condition on the initial datum.
A similar PDE with dispersion relation that is singular at low frequencies pops up in the linear water wave theory for the deep water case: the PDE would look something like$$ u_t-|\partial_x|^{-1/2}u=0. $$