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Optimal factors in Vladimir Markov's inequality in L2 norm

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Języki publikacji
EN
Abstrakty
EN
In this paper we discuss a problem of computation of constants in Vladimir Markov's type inequality in L2 norm on the interval [-1, 1].
Słowa kluczowe
Rocznik
Strony
64--73
Opis fizyczny
Bibliogr. 36 poz., tab.
Twórcy
autor
  • State Higher Vocational School in Tarnow, Mickiewicza 8, 33-100 Tarnów, Poland
autor
  • Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland
autor
  • State Higher Vocational School in Tarnow, Mickiewicza 8, 33-100 Tarnów, Poland
Bibliografia
  • [1] R.P. Agarwal, G.V. Milovanović, Extremal problems, inequalities and classical orthogonal polynomials, Appl. Math. Comput., 128 (2002), 151–166.
  • [2] D. Aleksov, G. Nikolov, A. Shadrin, On the Markov inequality in the L2-norm with the Gegenbauer weight, J. Approx. Theory 208 (2016), 9–20.
  • [3] D. Aleksov, G. Nikolov, Markov L2 inequality with the Gegenbauer weight, J. Approx. Theory 225 (2018), 224–241.
  • [4] G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyklopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge (1999).
  • [5] M. Baran, New approch to Markov inequality in Lp norms, Approximation Theory: in Memory of A.K. Varma (N.K. Govil and alt., ed.), Marcel Dekker, New York (1998), 75-85.
  • [6] M. Baran, L. Białas-Cież, B. Milówka, On the best exponent in Markov inequality, Potential Analysis, 38 (2) (2013), 635–651.
  • [7] B. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (2) (1982), 181-190.
  • [8] P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, Berlin, 1995, Graduate Texts in Mathematics 161.
  • [9] A. Böttcher, P. Dörfler, Wieighted Markov-type inequalities, norms of Volterra operators and zeros of Bessel functions, Math. Nachr. 283 (2010), 40–57.
  • [10] A. Böttcher, P. Dörfler, On the best constants in Markov-type inequalities involving Laguerre norms with different weights, Monatshefte f. Math. 161 (2010) 357–367.
  • [11] A. Böttcher, P. Dörfler, On the best constants in Markov-type inequalities involving Gegenbauer norms with different weights, Operators and Matrices 5 (2011), 261–272.
  • [12] Z. Ciesielski, On the A. A. Markov inequality for polynomials in the Lp case, in: ”Approximation theory”, Ed.: G. Anastassiou, pp., 257-262, Marcel Dekker, inc., New York, 1992.
  • [13] I.K. Daugavet, S.Z. Rafal’son, Certain inequalities of Markov-Nikolski type for algebraic polynomials, Vestnik Leningrad. Univ. 1 (1972), 15– 25 (Russian).
  • [14] D.K. Dimitrov, Markov Inequalities for Weight Functions of Chebyshev Type, J. Approx. Theory 83 (2) (1995), 175-181.
  • [15] P. Dörfler, New inequalities of Markov type, SIAM J. Math. Anal. (18), (1987), 490-494.
  • [16] P. Goetgheluck, On the Markov Inequality in Lp-Spaces, J. Approx. Theory 62 (2) (1990), 197-205.
  • [17] P.Yu. Glazyrina, The Sharp Markov-Nikol’skii Inequality for Algebraic Polynomials in The Spaces Lq and L0 on a Closed Interval, Mathematical Notes, 84 (1) (2007), 3-22.
  • [18] E. Hille, G. Szegö, J. Tamarkin, On some generalisation of a theorem of A. Markoff, Duke Math. J. 3 (1937), 729–739.
  • [19] A. Jonsson, Markov’s inequality and Zeros of Orthogonal Polynomials on Fractal Sets, J. Approx. Theory 78 (1994), 87–97.
  • [20] S.V. Konyagin, Estimates of derivatives of polynomials, Dokl. Acad. Nauk SSSR 243 (1978), 1116-1118 (Russian).
  • [21] G.K. Kristiansen, Some inequalities for algebraic and trigonometric polynomials, J. London Math. Soc. 20 (2) (1979), 300–314.
  • [22] A. Kroó, On the exact constant in the L2 Markov inequality, J. Approx. Theory 151 (2008), 208–211.
  • [23] G. Labelle, Concerning polynomials on the unit interval, Proc. Amer. Math. Soc. 20 (1969), 321-326.
  • [24] G.V. Milovanović, D.S. Mitrinović, T.M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientific , Singapore (1994).
  • [25] Q.I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press, Oxford (2002).
  • [26] E. Schmidt, Die asymptotische Bestimmung des Maximums des Integrals über das Quadrat der Ableitung eines normierten Polynoms, Sitzungsberichte der Preussischen Akademie, (1932), 287.
  • [27] E. Schmidt, Uber die nebst ihren Ableitungen orthogonalen Polynomen- ¨ systeme und das zugehörige Extremum, Math. Ann. 119 (1944), 165–204.
  • [28] L.F. Shampine, Some L2 Markoff inequalities, J. Res. Nat. Bur. Standards 69B (1965), 155–158.
  • [29] L.F. Shampine, An inequality of E. Schmidt, Duke Math. J. 33 (1966), 145–150.
  • [30] J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Verlag (2011).
  • [31] I.E. Simonov, Sharp Markov Brothers Type inequality in the Spaces Lp and L1 on a Closed Interval, Proceedings of the Steklov Institute of Mathematics, 277, Suppl. 1 (2012), S161-S170.
  • [32] G. Sroka, Constants in V.A. Markov’s inequality in LP norms, J. Approx. Theory 194 (2015), 27–34.
  • [33] E.M. Stein, Interpolation in polynomial classes and Markoff ’s inequality, Duke Math. J. 24 (1957), 467–476.
  • [34] G. Szegő, Orthogonal polynomials, American Mathematical Society Coloquium Publications 23, American Mathematical Society, Providence, RI, (2003).
  • [35] A.K. Varma, On Some Extremal Properties of Algebraic Polynomials, J. Approx. Theory 69 (1) (1992), 48–54.
  • [36] A. Zygmund, A remark on conjugate functions, Proceedings of the London Math. Soc. 34 (1932), 392-400.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-81e90e67-6e2a-421f-9fc4-056eebe2a0e3
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