JournalsjemsVol. 19, No. 1pp. 67–106

Bernstein inequalities with nondoubling weights

  • Andrii Bondarenko

    University of Trondheim, Norway
  • Sergey Tikhonov

    Centre de Recerca Matemática, Bellaterra (Barcelona), Spain
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We answer Totik's question on weighted Bernstein's inequalities by showing that

TnLp(ω)C(p,ω)nTnLp(ω),0<p,\|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 TnT_n and certain nondoubling weights ω\omega. Moreover, we find necessary conditions on ω\omega for Bernstein's inequality to hold. We also prove weighted Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.

Cite this article

Andrii Bondarenko, Sergey Tikhonov, Bernstein inequalities with nondoubling weights. J. Eur. Math. Soc. 19 (2017), no. 1, pp. 67–106

DOI 10.4171/JEMS/661