Consider the Fourier transform of $\exp(-z^k)$ where $k$ is a positive integer. As the function is analytic, I expect it to
have exponential decay at infinity. Is there some known theorem giving a quantitative estimate for that decay?
(Some variant of the Paley–Wiener theorem useful for this case?)
Off course if k=2 we get (up to a constants) the same function, but I don't expect an explicit formula for other values of k.
You have to restrict yourself to $k$ even, as for odd $k$ the function is not integrable and thus the Fourier transform does not exists (at least in a conventional sense).
So you are interested in the value of $$ F_n(k)=\int_{-\infty}^\infty e^{i k x - x^{2n}}\,dx $$ for $|k|\to \infty$. The saddle point $x_*$ is given by the solution of $$ \frac{d}{dx} (i k x - x^{2n}) = i k - 2n x^{2n-1} =0\,.$$ We obtain $$ x_*= c_1 k^{\frac{1}{2n-1}}\,.$$ So the dominating behavior of the integral is given by $$ F_n(k) \sim c_2 \exp\left(-c_3 k^{\frac{2n}{2n-1} } \right)$$ where $c_1,c_2,c_3$ are constants which can be determined by applying the method of stationary phase carefully.