What transform is this?

I need to compute the following transform numerically:

$F(\omega) = \int_{-\infty}^{\infty}f(x)e^{\omega(ix - g(x))}\text{d}x$

(note that if $g(x) = 0\ \forall\,x$ this reduces to the Fourier transform, which can be computed efficiently using FFT.)

Does this transform have a name, and is there any way to compute it efficiently when $f(x)$ and $g(x)$ are discretized?