On Vladimir Markov type inequality in L^p norms on the interval [-1; 1]

• Autorzy: Baran Mirosław , Ozorka Paweł

Identyfikatory

DOI

10.5604/01.3001.0013.7225

• Języki publikacji: EN

Abstrakty

EN

We prove inequality ||P(k)||Lp(-1;1)≤Bp||Tn(k)||Lp(-1;1)n^(2/p) ||P||Lp(-1;1); where Bp are constants independent of n = deg P with 1 ≤ p ≤ 2, which is sharp in the case k ≥ 3. A method presented in this note is based on a factorization of linear operator of k-th derivative throughout normed spaces of polynomial equipped with a Wiener type norm.

Słowa kluczowe

EN

Vladimir Markov type inequality; Lp-norm

PL

nierówności typu Władimira Markowa; norma Lp

• Wydawca: Państwowa Wyższa Szkoła Zawodowa w Tarnowie
• Czasopismo: Science, Technology and Innovation
• Rocznik: 2019
• Tom: Vol. 7, no. 4
• Strony: 9--12
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Baran Mirosław

• Faculty of Mathematics, Physics and Technical Science, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

autor

Ozorka Paweł

• Department of Mathematics, Faculty of Mathematical and Natural Sciences, University of Applied Sciences in Tarnow, Mickiewicza 8, 33-100 Tarnów, Poland

Bibliografia

• [1] 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.
• [2] M. Baran, L. Białas-Cież, Hölder continuity of the Green function and Markov brothers’ inequality, Constr. Approx. 40 (2014), no. 1, 121-140.
• [3] M. Baran, L. Białas-Cież, B. Milówka, On the best exponent in Markov inequality, Potential Analysis, 38 (2) (2013), 635–651.
• [4] M. Baran, A. Kowalska, P. Ozorka, Optimal factors in Vladimir Markov’s inequality in L2 norm, STI (2018).
• [5] M. Baran, B. Milówka, P. Ozorka, Markov’s property for k-th derivative, Ann. Polon. Math., 106 (2012), 31–40.
• [6] J. Bergh, J.Löfström, Interpolation Spaces. An Introduction, Springer Verlag, Berlin-Heidelberg-New York, (1976).
• [7] L. Białas-Cież, G. Sroka, Polynomial inequalities in Lp norms with generalized Jacobi weights, Math. Inequal. Appl. 22 (2019), no. 1, 261-274
• [8] P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, Berlin, 1995, Graduate Texts in Mathematics 161.
• [9] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190.
• [10] 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.
• [11] P. Goetgheluck, On the Markov Inequality in Lp-Spaces, J. Approx. Theory 62 (2) (1990), 197-205.
• [12] 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.
• [13] E. Hille, G. Szegö, J. Tamarkin, On some generalisation of a theorem of A. Markoff, Duke Math. J. 3 (1937), 729-739.
• [14] Y. Katznelson, An introduction to harmonic analysis, (Third. ed.), New York, (2004).
• [15] B. Milówka, Markov’s inequality and a generalized Plesniak condition, East J. Approx. 11 (2005), 291–300.
• [16] Q. Rahman, G. Schmeisser, Analytic theory of polynomials, Clarendon Press, (2002).
• [17] W. Rudin, Functional Analysis, McGraw-Hill Book
• [18] E. Schmidt, Die asymptotische Bestimmung des Maximums des Integrals über das Quadrat der Ableitung eines normierten Polynoms, Sitzungsberichte der Preussischen Akademie, (1932), 287.
• [19] I.E. Simonov, Sharp Markov Brothers Type inequality in the Spaces Lp and L1 on a Closed Interval, Proc. Steklov Inst. of Math., 277, Suppl. 1 (2012), S161-S170.
• [20] G. Sroka, Constants in Markov’s inequality in LP ([−1, 1]) norms for k – th derivative of an algebraic polynomial, J. Approx. Theory 194 (2015,) 27-34
• [21] E.M. Stein, Interpolation in polynomial classes and Markoff’s inequality, Duke Math. J. 24 (1957), 467-476.
• [22] A. Zygmund, A remark on conjugate functions, Proc. London Math. Soc. 34 (1932), 392–400.