Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials |
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| Authors: | A. I. Podvysotskaya |
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| Affiliation: | (1) Saratov State University, Astrakhanskaya Street 83, 410012 Saratov, Russia;(2) Russian Academy of Science, V.A. Kotel’nikov Institute of RadioEngineering and Electronics, Saratov Branch, Zelyonaya Str. 38, 410019 Saratov, Russia;; |
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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 ( n - 1 ) mathord | / | vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{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 | x | ? èk = 0[ n mathord | / | vphantom n 2 2 ] [ cosfrac2k + 12( n - 1 )p, cosfrac2k + 12np ] left| x right| in bigcupnolimits_{k = 0}^{left[ {{n mathord{left/{vphantom {n 2}} right.} 2}} right]} {left[ {cos frac{{2k + 1}}{{2left( {n - 1} right)}}pi, cos frac{{2k + 1}}{{2n}}pi } right]} . |
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