Conditions for symmetric and unimodal windows in both time and frequency domains After a lecture on harmonic analysis and time/frequency methods, I reconsidered the Gaussian kernel, defined in continuous time. 

It is unimodal and symmetric, and its continuous Fourier transform is another Gaussian, thus unimodal and symmetric (from Fourier transform of a Gaussian is not a Gaussian, but that's wrong!):



As a filter or a window, the Gaussian ensure a monotonic weight decay around its center. The interpretation in the frequency domain is similar: frequencies are monotonically attenuated away from a center frequency.

Such a property allows a straightforward interpretation in both the time and the frequency domain.


For continuous kernels, are there generic characterizations (necessary or sufficient conditions) under which  unimodal and symmetric windows also have a  unimodal and symmetric amplitude spectrum?
A sufficient condition is, that the distribution is spanned by zero-mean Gauss distributions with non-negative amplitude weights. The spanned space contains only symmetric and unimodal distributions and is invariant under a Fourier transform.