Extremal problems in de Branges spaces: the case of truncated and odd functions

TL;DR

This paper develops extremal one-sided approximation methods of exponential type for truncated and odd functions, extending previous work to broader classes with applications to periodic functions and utilizing advanced functional analysis techniques.

Contribution

It introduces new extremal approximation techniques for truncated and odd functions, generalizing prior results and applying reproducing kernel Hilbert space theory and interpolation methods.

Findings

Extended extremal approximation to broader classes of functions.

Provided periodic analogues with optimal trigonometric polynomial approximations.

Utilized advanced interpolation and kernel space techniques for approximation.

Abstract

In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the $L^1(\mathbb{R}, |E(x)|^{-2}dx)$-error, where $E$ is an arbitrary Hermite-Biehler entire function of bounded type in the upper half-plane. This extends the work of Holt and Vaaler [Duke Math. Journal 83 (1996), 203-247] for the signum function. We also provide periodic analogues of these results, finding optimal one-sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination. These extremal trigonometric polynomials optimize the $L^1(\mathbb{R}/\mathbb{Z}, d\vartheta)$-error, where $\vartheta$ is an arbitrary nontrivial measure on $\mathbb{R}/\mathbb{Z}$. The periodic results extend the work of Li and Vaaler [Indiana Univ. Math. J. 48, No. 1 (1999), 183-236], who considered this problem for the sawtooth function with respect to Jacobi measures. Our techniques are based on the theory of reproducing kernel Hilbert spaces (of entire functions and of polynomials) and on the construction of suitable interpolations with nodes at the zeros of Laguerre-P\'{o}lya functions.