Extremal Signatures

TL;DR

This paper characterizes extremal signatures for certain function spaces using Hermite-Biehler functions, enabling optimal approximation of functions like Gaussian and Poisson kernel in specific norms.

Contribution

It introduces a novel connection between Hermite-Biehler functions and extremal signatures, facilitating best approximation results in $L^1$-norm.

Findings

Sign of the product AB is an extremal signature for the space.

Allows explicit construction of best approximations for special functions.

Extends approximation theory using complex analysis techniques.

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 $A B$ 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).