Bernstein- and Markov-Type Inequalities for Trigonometric Polynomials on General Sets

Abstract

Bernstein- and Markov-type inequalities are discussed for the derivatives of trigonometric and algebraic polynomials on general subsets of the real axis and of the unit circle. It has recently been proved by A. Lukashov that the sharp Bernstein factor for trigonometric polynomials is the equilibrium density of the image of the set on the unit circle under the mapping t→eit. In this paper, Lukashov's theorem is extended to entire functions of exponential type using a result of Achieser and Levin. The asymptotically sharp Markov factors for trigonometric polynomials on several intervals is also found via the so-called T-sets of F. Peherstorfer and R. Steinbauer. This sharp Markov factor is again intimately connected with the equilibrium measure of the aforementioned image set.