Shape of Kaiser windowing function FFT response
The Kaiser window approximates the discrete prolate spheroidal sequence (DPSS) window which I think tends to a Gaussian function in an appropriate limit of large window and large $\alpha$, because the DPSS window concentrates energy both in time and frequency domain, as does the Gaussian function. The the zero-phase Kaiser window of length $N$ with shape parameter $\alpha$ is:

$$w_0(n) = \frac{I_0\left(\pi\alpha \sqrt{1-(\frac{2 n}{N-1})^2}\right)}{I_0(\pi\alpha)},$$

where $I_0$ is the zeroth-order modified Bessel function of the first kind. Substituting $n=\frac{(N-1)t}{2\sqrt{\alpha}}$ appears to normalize time so that the shape of the window does not change at large $\alpha$:

$$= \frac{I_0\left(\pi\sqrt{\alpha^2-\alpha t^2}\right)}{I_0(\pi\alpha)}$$

I was unable to calculate the limit $\alpha \rightarrow \infty$ symbolically but numerical testing indicates that at large $\alpha$ the window tends to a peak-normalized Gaussian function:

$$\approx e^{-\displaystyle\frac{\pi t^2}{2}}.$$

The maximum difference between the window and the Gaussian appears to be in the order of $\frac{1}{\alpha}$. The Fourier transform of a Gaussian function is a Gaussian function.