Optimal factors in Vladimir Markov’s inequality in L2 Norm

Authors

  • Mirosław Baran State Higher Vocational School in Tarnow, Poland
  • Agnieszka Kowalska Pedagogical University of Krakow, Poland
  • Paweł Ozorka State Higher Vocational School in Tarnow, Poland https://orcid.org/0000-0002-6424-9118

DOI:

https://doi.org/10.55225/sti.231

Keywords:

V. Markov’s inequality, L2 norms

Abstract

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].

Downloads

Download data is not yet available.

R.P. Agarwal, G.V. Milovanović, Extremal problems, inequalities and classical orthogonal polynomials, Appl. Math. Comput., 128 (2002), 151–166. DOI: https://doi.org/10.1016/S0096-3003(01)00070-4   Google Scholar

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. DOI: https://doi.org/10.1016/j.jat.2016.03.005   Google Scholar

D. Aleksov, G. Nikolov, Markov L2 inequality with the Gegenbauer weight, J. Approx. Theory 225 (2018), 224–241. DOI: https://doi.org/10.1016/j.jat.2017.10.008   Google Scholar

G.E. Andrews, R. Askey, R. Roy, Special Functions, Encyklopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge (1999).   Google Scholar

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.   Google Scholar

M. Baran, L. Bialas-Cie˙z, B. Mil´owka, On the best exponent in Markov inequality, Potential Analysis, 38 (2) (2013), 635–651. DOI: https://doi.org/10.1007/s11118-012-9290-0   Google Scholar

B. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (2) (1982), 181-190. DOI: https://doi.org/10.1016/0021-9045(82)90036-3   Google Scholar

P. Borwein, T. Erd´elyi, Polynomials and Polynomial Inequalities, Springer, Berlin, 1995, Graduate Texts in Mathematics 161. DOI: https://doi.org/10.1007/978-1-4612-0793-1   Google Scholar

A. Böttcher, P. D¨orfler, Wieighted Markov-type inequalities, norms of Volterra operators and zeros of Bessel functions, Math. Nachr. 283 (2010), 40–57. DOI: https://doi.org/10.1002/mana.200810274   Google Scholar

A. Böttcher, P. D¨orfler, On the best constants in Markov-type inequalities involving Laguerre norms with different weights, Monatshefte f. Math. 161 (2010) 357–367. DOI: https://doi.org/10.1007/s00605-009-0187-y   Google Scholar

A. B¨ottcher, P. D¨orfler, On the best constants in Markov-type inequalities involving Gegenbauer norms with different weights, Operators and Matrices 5 (2011), 261–272. DOI: https://doi.org/10.7153/oam-05-18   Google Scholar

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.   Google Scholar

I. K. Daugavet, S. Z. Rafal’son, Certain inequalities of Markov-Nikolski type for algebraic polynomials, Vestnik Leningrad. Univ. 1 (1972), 15–25 (Russian).   Google Scholar

D. K. Dimitrov, Markov Inequalities for Weight Functions of Chebyshev Type, J. Approx. Theory 83 (2) (1995), 175-181. DOI: https://doi.org/10.1006/jath.1995.1115   Google Scholar

P. Dörfler, New inequalities of Markov type, SIAM J. Math. Anal. (18), (1987), 490-494. DOI: https://doi.org/10.1137/0518039   Google Scholar

P. Goetgheluck, On the Markov Inequality in Lp-Spaces, J. Approx. Theory 62 (2) (1990), 197-205. DOI: https://doi.org/10.1016/0021-9045(90)90032-L   Google Scholar

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. DOI: https://doi.org/10.1134/S0001434608070018   Google Scholar

E. Hille, G. Szegö, J. Tamarkin, On some generalisation of a theorem of A. Markoff, Duke Math. J. 3 (1937), 729–739. DOI: https://doi.org/10.1215/S0012-7094-37-00361-2   Google Scholar

A. Jonsson, Markov’s inequality and Zeros of Orthogonal Polynomials on Fractal Sets, J. Approx. Theory 78 (1994), 87–97. DOI: https://doi.org/10.1006/jath.1994.1062   Google Scholar

S. V. Konyagin, Estimates of derivatives of polynomials, Dokl. Acad. Nauk SSSR 243 (1978), 1116-1118 (Russian).   Google Scholar

G. K. Kristiansen, Some inequalities for algebraic and trigonometric polynomials, J. London Math. Soc. 20 (2) (1979), 300–314. DOI: https://doi.org/10.1112/jlms/s2-20.2.300   Google Scholar

A. Kroó, On the exact constant in the L2 Markov inequality, J. Approx. Theory 151 (2008), 208–211. DOI: https://doi.org/10.1016/j.jat.2007.09.006   Google Scholar

G. Labelle, Concerning polynomials on the unit interval, Proc. Amer. Math. Soc. 20 (1969), 321-326. DOI: https://doi.org/10.1090/S0002-9939-1969-0236568-0   Google Scholar

G. V. Milovanovi´c, D.S. Mitrinović, T. M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientific , Singapore (1994). DOI: https://doi.org/10.1142/1284   Google Scholar

Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press, Oxford (2002).   Google Scholar

E. Schmidt, Die asymptotische Bestimmung des Maximums des Integrals ¨uber das Quadrat der Ableitung eines normierten Polynoms, Sitzungsberichte der Preussischen Akademie, (1932), 287.   Google Scholar

E. Schmidt, Uber die nebst ihren Ableitungen orthogonalen Polynomen-systeme und das zugeh¨orige Extremum, Math. Ann. 119 (1944), 165–204. DOI: https://doi.org/10.1007/BF01563739   Google Scholar

L. F. Shampine, Some L2 Markoff inequalities, J. Res. Nat. Bur. Standards 69B (1965), 155–158. DOI: https://doi.org/10.6028/jres.069B.017   Google Scholar

L. F. Shampine, An inequality of E. Schmidt, Duke Math. J. 33 (1966), 145–150. DOI: https://doi.org/10.1215/S0012-7094-66-03319-9   Google Scholar

J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Verlag (2011). DOI: https://doi.org/10.1007/978-3-540-71041-7   Google Scholar

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. DOI: https://doi.org/10.1134/S0081543812050161   Google Scholar

G. Sroka, Constants in V.A. Markov’s inequality in LP norms, J. Approx. Theory 194 (2015), 27–34. DOI: https://doi.org/10.1016/j.jat.2014.12.010   Google Scholar

E. M. Stein, Interpolation in polynomial classes and Markoff ’s inequality, Duke Math. J. 24 (1957), 467–476. DOI: https://doi.org/10.1215/S0012-7094-57-02453-5   Google Scholar

G. Szegö, Orthogonal polynomials, American Mathematical Society Coloquium Publications 23, American Mathematical Society, Providence, RI, (2003).   Google Scholar

A. K. Varma, On Some Extremal Properties of Algebraic Polynomials, J. Approx. Theory 69 (1) (1992), 48–54. DOI: https://doi.org/10.1016/0021-9045(92)90048-S   Google Scholar

A. Zygmund, A remark on conjugate functions, Proceedings of the London Math. Soc. 34 (1932), 392-400. DOI: https://doi.org/10.1112/plms/s2-34.1.392   Google Scholar

Downloads

Published

2018-06-28

How to Cite

Baran, M., Kowalska, A., & Ozorka, P. (2018). Optimal factors in Vladimir Markov’s inequality in L2 Norm. Science, Technology and Innovation, 2(1), 64–73. https://doi.org/10.55225/sti.231

Issue

Section

Original articles