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Arestov’s theorems on Bernstein’s inequality. (English) Zbl 1439.41007

V. V. Arestov proved in [Math. USSR, Izv. 18, 1–18 (1982; Zbl 0538.42001)] a Bernstein type inequality for trigonometric polynomials in \( L_p, p\geq 0. \) The author, following Arestov’s \(\Lambda\)-method from the cited paper, gives a simple, elementary, and at least partially new proof of this Arestov’s result. The crucial observation is that Boyd’s approach to prove Mahler’s inequality for algebraic polynomials can be extended to all trigonometric polynomials. Section 4 contains a clearly presented history of the topic.

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)

Citations:

Zbl 0538.42001
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References:

[1] Arestov, V. V., On inequalities of S.N Bernstein for algebraic and trigonometric polynomials, Soviet Math. Dokl., 20, 3, 600-603 (1979) · Zbl 0433.41004
[2] Arestov, V. V., On integral inequalities for trigonometric polynomials and their derivatives Izv, AN SSSR Ser. Mat., 45, 1, 3-22 (1981), (Russian), translation in Math. USSR Izv. 18 (1982), no. 1, 1-17 · Zbl 0517.42001
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[12] Nevai, P., The True Story of \(n\) vs \(2 n\) in the Bernstein Inequality (2019), in preparation
[13] Nevai, P.; The Anonymous Referee, J. E., The Bernstein inequality and the Schur inequality are equivalent, J. Approx. Theory, 182, 103-109 (2014) · Zbl 1290.41007
[14] Pólya, G.; Szegő, G., Problems and Theorems in Analysis II (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0311.00002
[15] Queffélec, H.; Zarouf, R., On Bernstein’s inequality for polynomials, Anal. Math. Phys. (2019), in press · Zbl 1432.42002
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