  We prove that all functions obeying the Kramers-Kronig relations can be
approximated as superpositions of Lorentzian functions, to any precision. As a
result, the typical text-book analysis of dielectric dispersion response
functions in terms of Lorentzians may be viewed as encompassing the whole class
of causal functions. A further consequence is that Lorentzian resonances may be
viewed as possible building blocks for engineering any desired metamaterial
response, for example by use of split ring resonators of different parameters.
Two example functions, far from typical Lorentzian resonance behavior, are
expressed in terms of Lorentzian superpositions: A steep dispersion medium that
achieves large negative susceptibility with arbitrarily low loss/gain, and an
optimal realization of a perfect lens over a bandwidth. Error bounds are
derived for the approximation.
