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?