The perfect lens on a finite bandwidth

The resolution associated with the so-called perfect lens of thickness $d$ is $-2\pi d\mathrm{/}\ln (\mathrm{\mid }\chi +2\mathrm{\mid }\mathrm{/}2)$. Here the susceptibility $\chi$ is a Hermitian function in $H^2$ of the upper half-plane, i.e., a $H^2$ function satisfying $\chi (-\omega )=\overline{\chi (\omega )}$. An additional requirement is that the imaginary part of $\chi$ be nonnegative for nonnegative arguments. Given an interval $I$ on the positive half-axis, we compute the distance in $L^{\mathrm{\infty }}(I)$ from a negative constant to this class of functions. This result gives a surprisingly simple and explicit formula for the optimal resolution of the perfect lens on a finite bandwidth.