the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part.
This often comes up in physics, where it can be used to related resonances and absorption. What one usually finds there is the following: Where the imaginary part has a peak, the real part goes through zero.
Is this a general rule?
And are there more general statements possible? For Fourier transforms, for example, I know the statement that a peak with width $\Delta$ in time domain corresponds to a peak with width $1/\Delta$ (missing some factors $\pi$, I am sure...) in frequency domain.
Is there some rule of thumb that tells me how the Hilbert transform of a function with finite support (e.g. with a bandwidth $W$) looks like, approximately?