The sharp Remez-type inequality for even trigonometric polynomials on the period

TL;DR

This paper establishes sharp Remez-type inequalities for even and general trigonometric polynomials on the period, relating maximum modulus to measure constraints and Chebyshev polynomials.

Contribution

It provides the first sharp bounds for the maximum modulus of trigonometric polynomials under measure constraints, extending classical Remez inequalities.

Findings

Proves sharp bounds involving Chebyshev polynomials for even trigonometric polynomials.

Extends Remez-type inequalities to general trigonometric polynomials with measure constraints.

Establishes the inequalities as optimal.

Abstract

We prove that $$\max_{t \in [-\pi,\pi]}{|Q(t)|} \leq T_{2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{2n} + (\sec(s/4) - \tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0,2\pi)\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \subset {\Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(\cos t) = \cos(2nt)$ for $t \in {\Bbb R}$. This inequality is sharp. We also prove that $$\max_{t \in [-\pi,\pi]}{|Q(t)|} \leq T_{2n}(\sec(s/2)) = \frac 12 ((\sec(s/2) + \tan(s/2))^{2n} + (\sec(s/2) - \tan(s/2))^{2n})$$ for every trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0,\pi)\,.$$