Some extremal functions in Fourier analysis, II

TL;DR

This paper develops extremal functions of exponential type for specific even functions, providing applications in inequalities, polynomial bounds, and Hermitian forms within Fourier analysis.

Contribution

It introduces new extremal majorants and minorants for functions like log|x| and |x|^α, including periodic versions with trigonometric polynomials.

Findings

Derived optimal estimates for Hermitian forms.

Established periodic extremal functions with bounded degree.

Provided an Erdős-Turán-type inequality for polynomial root sums.

Abstract

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erd\"{o}s-Tur\'{a}n-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.