Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

TL;DR

This paper constructs optimal trigonometric polynomials to approximate Bernoulli periodic functions, providing bounds, inequalities, and applications in harmonic analysis.

Contribution

It introduces the first optimal trigonometric polynomial approximations for Bernoulli periodic functions, extending Littmann's work and generalizing Vaaler's results.

Findings

Optimal approximations in $L^1$ norm achieved

Erdős-Turán-type inequalities derived

Bounds for Hermitian forms established

Abstract

We obtain optimal trigonometric polynomials of a given degree $N$ that majorize, minorize and approximate in $L^1(\mathbb{R}/\mathbb{Z})$ the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann that generalize a paper of J. Vaaler. As applications we provide the corresponding Erd\"{o}s-Tur\'{a}n-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.