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Note on a conjecture about Bernstein type inequalities for multivariate polynomials

Об одной гипотеэе относительно неравенств типа Бернштейна для многочленов от нескольких переменных

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Abstract

There are fine extensions of the univariate Bernstein-Szegő inequality for multivariate polynomials considered on a convex domain K. The current one estimates the gradient of the polynomial P at a point x ∈ K by constant times degree, ‖P‖ C(K) and a geometrical factor. The best constant is within [2, 2√2]. In this note we disprove the conjecture (based on some particular cases) that the best constant is 2.

Abstract

Существуют раэные виды обобщений одномерного неравенства Бернштейна-Сегё для многочленов от нескольких переменных на выпуклой области K. Мы рассматриваем оценку градиента многочлена P в x ∈ K, которая имеет вид проиэведения константы, степени многочлена, ‖P‖ C (K) и некоторого множителя, который эависит от геометрии. Иэвестно, что для неравенств такого вида наилушая константа принадлежит отреэку [2, 2 √2]. В зтой работе мы покаэываем, что гипотеэа (основанная на рассмотрении некоторых частных случаев) о том, что наилучшая константа равна 2, не оправдывается.

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Authors and Affiliations

  1. Department of Mathematics, University of Sofia, Boulevard James Boucher 5, 1164, Sofia, Bulgaria

    Nikola Naidenov

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  1. Nikola Naidenov
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This study is supported by the research project “Approximation Theory” between the Bulgarian Academy of Sciences and the Hungarian Academy of Sciences.

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Naidenov, N. Note on a conjecture about Bernstein type inequalities for multivariate polynomials. Anal Math 33, 55–62 (2007). https://doi.org/10.1007/s10474-007-0105-2

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  • DOI: https://doi.org/10.1007/s10474-007-0105-2

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