Bernstein inequalities with nondoubling weights

Abstract

We answer Totik's question on weighted Bernstein's inequalities by showing that

holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $\omega$. Moreover, we find necessary conditions on $\omega$ for Bernstein's inequality to hold. We also prove weighted Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.