Extremal Signatures

Abstract

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 $AB$ 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).