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Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight

Authors: A. I. Aptekarev, A. Draux, V. A. Kalyagin and D. N. Tulyakov
Journal: Proc. Amer. Math. Soc. 143 (2015), 3847-3862
MSC (2010): Primary 41A20, 33C45, 42C05; Secondary 47B99, 30B70
Published electronically: May 4, 2015
MathSciNet review: 3359576
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Abstract: The classical A. Markov inequality establishes a relation between the maximum modulus or the $ L^{\infty }\left ([-1,1]\right )$ norm of a polynomial $ Q_{n}$ and of its derivative: $ \Vert Q'_{n}\Vert\leqslant M_{n} n^{2}\Vert Q_{n}\Vert$, where the constant $ M_{n}=1$ is sharp. The limiting behavior of the sharp constants $ M_{n}$ for this inequality, considered in the space $ L^{2}\left ([-1,1], w^{(\alpha ,\beta )}\right )$ with respect to the classical Jacobi weight $ w^{(\alpha ,\beta )}(x):=(1-x)^{\alpha }(x+1)^{\beta }$, is studied. We prove that, under the condition $ \vert\alpha - \beta \vert < 4 $, the limit is $ \lim _{n \to \infty } M_{n} = 1/(2 j_{\nu })$ where $ j_{\nu }$ is the smallest zero of the Bessel function $ J_{\nu }(x)$ and $ 2 \nu =$$ \mbox {min}(\alpha , \beta ) - 1$.

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  • [1] A.A. Markov, On one question of D.I. Mendeleev, Izvestiya Peterburg Akademii Nauk, 62, (1889), 1-24 (in Russian).
  • [2] S.N. Bernshtein, On the best approximation of the continuous functions by means of polynomials with fixed degree, Soobsheniya Kharkovskogo Matem. Obshestva, (1912), (in Russian).
  • [3] N.I. Akhiezer, Lectures in Approximation Theory, 3rd ed., Nauka, Moscow 1972; English transl. Theory of Approximation, Dover Publ., New York 1992.
  • [4] G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187
  • [5] M. Abramowitz and I. Stegun (Editors), Handbook of Mathematics Functions, National Bureau of Standards, 10-th Edition, NY, 1972.
  • [6] F. Gantmacher Theory of matrix, v.1, Am. Math. Soc., 2000.
  • [7] Frank Bowman, Introduction to Bessel functions, Dover Publications Inc., New York, 1958. MR 0097539
  • [8] A. I. Aptekarev, A. Dro, and V. A. Kalyagin, On the asymptotics of exact constants in Markov-Bernstein inequalities in integral metrics with classical weight, Uspekhi Mat. Nauk 55 (2000), no. 1(331), 173–174 (Russian); English transl., Russian Math. Surveys 55 (2000), no. 1, 163–165. MR 1751821,
  • [9] André Draux and Charaf Elhami, On the positivity of some bilinear functionals in Sobolev spaces, J. Comput. Appl. Math. 106 (1999), no. 2, 203–243. MR 1696408,
  • [10] Alexandre I. Aptekarev, André Draux, and Dmitrii Toulyakov, Discrete spectra of certain co-recursive Pollaczek polynomials and its applications, Comput. Methods Funct. Theory 2 (2002), no. 2, [On table of contents: 2004], 519–537. MR 2038136,
  • [11] André Draux and Valeri Kaliaguine, Markov-Bernstein inequalities for generalized Hermite weight, East J. Approx. 12 (2006), no. 1, 1–23. MR 2294666
  • [12] A. I. Aptekarev, Asymptotics of orthogonal polynomials in a neighborhood of endpoints of the interval of orthogonality, Mat. Sb. 183 (1992), no. 5, 43–62 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 76 (1993), no. 1, 35–50. MR 1184309,
  • [13] D. N. Tulyakov, On the local asymptotics of the ratio of orthogonal polynomials in a neighborhood of an extreme point of the support of the orthogonality measure, Mat. Sb. 192 (2001), no. 2, 139–160 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 1-2, 299–321. MR 1835990,
  • [14] D.N. Tulyakov, Difference equations having bases with powerlike growth which are perturbed by a spectral parameter, Matem. sb., 200:5 (2009), 129-158; English transl. in Sb. Math., 200:5 (2009), 753-781.
  • [15] G. Szegő, Orthogonal polynomials, rev. ed., Amer. Math. Soc, Providence, RI, 1959.

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Additional Information

A. I. Aptekarev
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia and Moscow State University, Moscow, Russia

A. Draux
Affiliation: INSA, Rouen, France

V. A. Kalyagin
Affiliation: National Research University Higher School of Economics, Nizhny Novgorod, Russia

D. N. Tulyakov
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

Received by editor(s): June 11, 2013
Received by editor(s) in revised form: February 24, 2014, and April 18, 2014
Published electronically: May 4, 2015
Additional Notes: The first and fourth authors were partly supported by the program N1 of DMS RAS and grants RFBR-13-01-12430, RFBR-14-01-00604. The third author was partly supported by the Scientific Schools program - 2900.2014.1. The paper was finished while the first author visited INSA, Rouen, France.
Communicated by: W. Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

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