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 $0$. 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 case less rigorous.
SteveH
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