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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?

Tanks, Lagerbaer

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Where the imaginary part has a peak, the real part... is there something missing? – Jonas Teuwen Jun 23 '11 at 23:51
Yes... funny enough, there was something missing. – Lagerbaer Jun 24 '11 at 15:35

Never heard of the Kramers-Kronig relations and so I looked it up. It relates the real and imaginary parts of an analytic function on the upper half plane that satisfies certain growth conditions. This is a big area in complex analysis and there are many results. For example, in the case of a function with compact support, its Hilbert transform can never have compact support, or even vanish on a set of measure greater $0$. Many books on analytic functions (especially ones on $H^p$ spaces and bounded analytic functions) cover this topic. Some books in signal processing also cover this but from a different perspective, and in most cases less rigorous.

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Well, the Kramers-Kronig relations are well-known by those who know [them] well :) – t.b. Jun 24 '11 at 0:55

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