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

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 $$m(\{t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0,\pi)\,.$$