Bernstein inequalities with nondoubling weights

TL;DR

This paper establishes weighted Bernstein inequalities for trigonometric polynomials with nondoubling weights, providing conditions on weights and extending classical inequalities like Bernstein-Markov, Remez, and Nikolskii.

Contribution

It proves Bernstein inequalities for nondoubling weights and identifies necessary conditions on weights, extending classical polynomial inequalities.

Findings

Bernstein inequalities hold for certain nondoubling weights

Necessary conditions on weights for Bernstein inequalities

Weighted Bernstein-Markov, Remez, and Nikolskii inequalities proved

Abstract

We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(\omega)} \le C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad 0<p\le \infty, $$ 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 Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.