I'm trying to find an explicit result for the following Fourier transform:
$$\mathcal{F}\left[e^{if(x)}\right](k)=\int_{\mathbb{R}^n} e^{if(x)}e^{-ik\cdot x} dx$$
So far I could come up only with a symbolic solution which doesn't help me a lot. I report it below in case it were of any use to somebody.
Let's expand $e^{if(x)}$ as:
$$e^{if(x)} = \sum_{n=0}^{\infty} \frac{i^n}{n!}(f(x))^n$$
Applying the Fourier transform to the sum one gets:
$$ \sum_{n=0}^{\infty} \frac{i^n}{n!}\mathcal{F}[f^n](k) = \delta(k) + \sum_{n=1}^{\infty} \frac{i^n}{n!}(\underbrace{\tilde f*\tilde f*\dots*\tilde f}_{n})(k)\equiv (e^{i\tilde f *}f)(k)$$