For a problem I am trying to solve, I am required to calculate the Fourier transform of $e^{i f(x)}$. However, it so happens that I do not have access to $f(x)$ but only to its Fourier transform coefficients $\mathcal{F}f(\omega)$. Are you aware of any way to define $\mathcal{F}(e^{i f(x)})$ in terms of $\mathcal{F}f$?
Thank you all in advance!