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

Abstract

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

This is submitted to Springer volume "Topics in Classic and Modern Analysis. In memory of Yingkang Hu", Applied and Numerical Harmonic Analysis series