Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?
Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?


Have a look at the Fourier Transfrom of a Gaussian Signal.

$$ \mathcal{F}_{x} \left\{ {e}^{-a {x}^{2} } \right\} \left( \omega \right) = \sqrt{\frac{\pi}{a}} {e}^{- {\pi}^{2} \frac{ {\omega}^{2} }{a} } $$

First, Gaussian Signal stays Gaussian under Fourier Transform.

As you can see, the parameter which multiplies the variable is inverted.
Let's say $ a = 5 $, then it means that in time we will have very sharp and thin Gaussian while in frequency we will have very smooth and wide Gaussian.

This is related to other property of Forier Transform.
In simple words, what's thin on Time / Spatial Domain is Wide on Frequency Doman and vise versa.
This is basically the Uncertainty Principle of Fourier Transform.