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On a conjecture of S. Bernstein in approximation theory. (English. Russian original) Zbl 0661.41005

Math. USSR, Sb. 57, 547-560 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 4, 535-548 (1986).
Denote by \(E_{2n}(| x|)\) the best approximation on the interval \([-1,+1]\) of the function \(| x|\) by algebraic polynomials of degree at most 2n. In 1914, S. N. Bernstein proved that there exists a positive constant \(\beta\) such that \(\lim_{n\to \infty}(2n E_{2n}(| x|))=\beta\) and showed that \(0.278<\beta <0.286\). Observing that \(\beta\) is closed (with two exact decimal digits) to 1/(2\(\sqrt{\pi})\), Bernstein conjectured that \(\beta =1/(2\sqrt{\pi})\). Using Richardson extrapolation method and exploring some ideas contained in the original paper of Bernstein, the authors solve in the negative this problem by evaluating the constant \(\beta\) with 50 exact decimal digits. A new conjecture, concerning an asymptotic evaluation for \(\beta\), is proposed.
Reviewer: S.Cobzaş

MSC:

41A10 Approximation by polynomials
41A25 Rate of convergence, degree of approximation
41A44 Best constants in approximation theory
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