Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

Abstract

Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below.

Theorem

Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that

$$\smallint _{ - \omega }^\omega \left| {T'_n (t)} \right|^p W(t)(\omega /n + \sqrt {\omega ^2 - t^2 )} ^p dt \leqslant Cn^p \smallint _{ - \omega }^\omega \left| {T_n (t)} \right|^p W(t)dt$$

holds for every T _n ∈ T _n , where T _n denotes the class of all real trigonometric polynomials of degree at most n .