Let $f(x) $ be a Schwartz function. Now, define the generalized Fourier transform of $f(x)$ as follows.
$$ F(\alpha):= \int f(x)\, e^{i\alpha \psi(x)}\,\mathrm dx $$
Where $\psi$ is continuously differentiable, but $\psi$ is not constant over any subinterval of $\mathbb{R}$.
I am wondering if $F(\alpha)$ is also a Schwartz function.
Thank you in advance for your help.