  Kramers-Kronig relations are extended to dielectric functions
$\varepsilon(q,\omega)$ that depend not only on the angular frequency $\omega$
  but on the wave number $q$ as well. This implies extending the notion of
causality commonly used in the theory of Kramers-Kronig relations
  to include the fact that signals cannot propagate faster than light in vacuo.
\\ The results derived here also apply to general theories of
  isotropic linear response in which the response function depends on both the
wave number and the frequency.
