Inequalities for algebraic polynomials on an ellipse

Tatiana M. Nikiforova<sup>ab

a</sup> Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci $\pm 1$ of the derivative of an algebraic polynomial with real coefficients normalized on the segment $[- 1,1]$.

Keywords: polynomial, Chebyshev polynomials, ellipse, segment, derivative of a polynomial, uniform norm.

Language: English

DOI: 10.15826/umj.2020.2.009

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