Integral transform that resembles the Fourier transform.

Question

I would like to take the following integral.

$$ F(k):= \int_{\mathbb{R}} e^{ikf(x)} g(x) dx$$

Where $g(x) \in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$ and $f(x)$ is an arbitrary fucntion.

Questions

1. Are there general methods for computing such a transform? For example, if $f(x):= x^{n}$ is anything known? More generally for $f(x) : = \sum_{n=0}^{\infty} a_{n}x^{n}$ is anything known? Any nice formulas etc.

2. Due to the Riemann-Lebesgue lemma we know that when $f(x):= x$, $F(k) \rightarrow 0$ as $|k| \rightarrow \infty$. Is there a similar result for certain families of functions $f(x)$?