Extremal Signatures

Let $E=A-iB$ be a Hermite-Biehler entire function of exponential type $\tau \mathrm{/}2$ where $A$ and $B$ are real entire, and consider $d\mu(x) = dx/|E(x)|^2$\mathrm{.}Weshowthatthesignoftheproduct$A B$ 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).