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Extremal Signatures

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