Trigonometric polynomials deviating the least from zero in measure and related problems

TL;DR

This paper addresses the problem of finding trigonometric and algebraic polynomials that deviate the least from zero under specific measure and uniform deviation criteria, providing explicit solutions and minimal values.

Contribution

It introduces solutions for minimal deviation problems of trigonometric and algebraic polynomials with fixed leading harmonic or zeros on the circle, advancing approximation theory.

Findings

Solved the least measure deviation problem for trigonometric polynomials with fixed harmonic.

Determined minimal uniform deviation for polynomials on compact sets with fixed measure.

Provided solutions for algebraic polynomials with zeros on the circle.

Abstract

We give a solution of the problem on trigonometric polynomials $f_n$ with the given leading harmonic $y\cos nt$ that deviate the least from zero in measure, more precisely, with respect to the functional $\mu(f_n)=mes\{t\in[0,2\pi]: |f_n(t)|\ge 1\}$. For trigonometric polynomials with a fixed leading harmonic, we consider the least uniform deviation from zero on a compact set and find the minimal value of the deviation over compact subsets of the torus that have a given measure. We give a solution of a similar problem on the unit circle for algebraic polynomials with zeros on the circle.