Abstract
The problem of the greatest possible absolute value of the kth derivative of an algebraic polynomial of order n > k with real coefficients at a given point of the complex plane is considered. It is assumed that the polynomial is bounded by 1 on the interval [-1,1]. It is shown that the solution is attained for the polynomial κ · Tσ, where Tσ is one of the Zolotarev or Chebyshev polynomials and κ is a number.
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References
V. M. Tikhomirov, Some Questions in Approximation Theory (Izd. Moskov. Univ., Moscow, 1976) [in Russian].
N. I. Akhiezer, Lectures in the Theory of Approximation (Nauka, Moscow, 1965) [in Russian].
G. G. Magaril-Il'yaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications (Editorial URSS, Moscow, 2000) [in Russian].
V. B. Demidovich, G. G. Magaril-Il'yaev, and V. M. Tikhomirov, “Extremal problems for linear functionals on the Tchebycheff spaces,” Fundam. Prikl. Mat. 11 (2), 87–100 (2005) [J. Math. Sci. (New York) 142 (2), 1923–1932(2007)].
R. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976; Mir, Moscow, 1979).
Funding
This work was supported by the Russian Foundation for Basic Research under grants 17-01-00649, 16-01-00295, and 17-01-00809.
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 4, pp. 543–548.
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Kochurov, A.S., Tikhomirov, V.M. On Extrapolation of Polynomials with Real Coefficients to the Complex Plane. Math Notes 106, 572–576 (2019). https://doi.org/10.1134/S0001434619090256
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DOI: https://doi.org/10.1134/S0001434619090256