  Let $E= A - iB$ be a Hermite-Biehler entire function of exponential type
$\tau/2$ where $A$ and $B$ are real entire, and consider $d\mu(x) =
dx/|E(x)|^2$. We show that the sign of the product $A B$ is an extremal
signature for the space of functions of exponential type $\tau$ with respect to
the norm of $L^1(\mu)$. This allows us to find best approximations by entire
functions of exponential type $\tau$ in $L^1(\mu)$-norm to certain special
functions (e.g., the Gaussian and the Poisson kernel).
