The Sharp Remez-Type Inequality for Even Trigonometric Polynomials on the Period

Abstract

We prove that

$$\displaystyle \max _{t \in [-\pi ,\pi ]}{|Q(t)|} \leq T_{2n}(\sec {}(s/4)) = \frac 12 ((\sec {}(s/4) + \tan {}(s/4))^{2n} + (\sec {}(s/4) - \tan {}(s/4))^{2n})$$

for every even trigonometric polynomial Q of degree at most n with complex coefficients satisfying

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

where m(A) denotes the Lebesgue measure of a measurable set $A \subset {\mathbb {R}}$ and T _2n is the Chebyshev 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

$$\displaystyle\max _{t \in [-\pi ,\pi ]}{|Q(t)|} \leq T_{2n}(\sec {}(s/2)) = \frac 12 ((\sec {}(s/2) + \tan {}(s/2))^{2n} + (\sec {}(s/2) - \tan {}(s/2))^{2n})$$

for every trigonometric polynomial Q of degree at most n with complex coefficients satisfying

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