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On Bernstein- and Markov-type inequalities S. I. Kalmykovabc a School of Mathematical Sciences, Shanghai Jiao Tong University b Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow c Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok |
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Abstract: Polynomial inequalities have various applications. For example, in approximation theory they are fundamental in establishing converse results, i.e., when one deduces smoothness from a given rate of approximation (see e.g. [1, p. 241]). In this talk we discuss classical Bernstein- and Markov-type inequalities for polynomials and rational functions as well as their recent generalizations. Mainly, we are interested in the results obtained with the help of potential theory and geometric function theory of a complex variable (for details see the surveys [2] and [3]). Key tools of proofs will be also considered. This is based joint work with V. Dubinin, B. Nagy and V. Totik. References [1] Borwein P. Erdélyi T., Polynomials and polynomial inequalities. Graduate Texts in Mathematics, 161. Springer-Verlag, New York, 1995. [2] Dubinin V. N., “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67 (4): 599–684, 2012. [3] Kalmykov S., Nagy B, Totik V., “Bernstein- and Markov-type inequalities”, Surveys in Approximation Theory, 9: 1–17, 2021. |
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