Bernstein inequalities with nondoubling weights

Abstract

We answer Totik&#39;s question on weighted Bernstein&#39;s inequalities showing that $$ \|T_n&#39;\|_{L_p(ω)} \le C(p,ω)\, {n}\,\|T_n\|_{L_p(ω)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $ω$. Moreover, we find necessary conditions on $ω$ for Bernstein&#39;s inequality to hold. We also prove weighted Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.