Abstract
We answer Totik's question on weighted Bernstein's inequalities by showing that \|T_n'\|_{L_p(\omega)} \leq C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad 0 < p \leq \infty, 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.
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