A Gaussian normal distribution "bell" on the time axis, is a normal on the frequency axis (by spectral transform). However, normal is essentially a cosine pulse until its elbow and tail; a pulse tail of zero is an overlap of sinc pairs in transform; spectral tail runout depends greatly on temporal tail runout:
A sharper elbow, flatter but noisier historic tail, may exist.
. . .
A premise discovery under the title,