2017
Том 69
№ 6

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Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

Podvysotskaya A. I.

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Abstract

We prove that $\max |p′(x)|$, where $p$ runs over the set of all algebraic polynomials of degree not higher than $n ≥ 3$ bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all $x ∈ (−1, 1)$ such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \).

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 5, pp 847-853.

Citation Example: Podvysotskaya A. I. Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials // Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 711-715.

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