On the Markov Inequality in the $$L_2$$ L 2 -Norm with the Gegenbauer Weight

Springer Science and Business Media LLC - Tập 49 Số 1 - Trang 1-27 - 2019
Geno Nikolov1, Alexei Shadrin2
1Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
2Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK

Tóm tắt

Từ khóa


Tài liệu tham khảo

Aleksov, D., Nikolov, G., Shadrin, A.: On the Markov inequality in the $$L_2$$ L 2 norm with the Gegenbauer weight. J. Approx. Theory 208, 9–20 (2016)

Bojanov, B.: Markov-type inequalities for polynomials and splines. In: Chui, C.K., Schumaker, L.L., Stoeckler, J. (eds.) Approximation Theory X. Abstract and Classical Analysis, pp. 31–90. Vanderbilt University Press, Vanderbilt (2002)

Böttcher, A.: Best constants for Markov type inequalities in Hilbert space norms. In: Recent Trends in Analysis, Proceedings of the Conference in Honor of Nikolai Nikolski, Bordeaux 2011, pp. 73–83, Theta, Bucharest (2013)

Böttcher, A., Dörfler, P.: Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions. Math. Nachr. 283, 357–367 (2010)

Böttcher, A., Dörfler, P.: On the best constant in Markov-type inequalities involving Gegenbauer norms with different weights. Oper. Matr. 161, 40–57 (2010)

Böttcher, A., Dörfler, P.: On the best constant in Markov-type inequalities involving Laguerre norms with different weights. Monatsh. Math. 5, 261–272 (2011)

Dörfler, P.: New inequalities of Markov type. SIAM J. Math. Anal. 18, 490–494 (1987)

Dörfler, P.: Asymptotics of the best constant in a certain Markov-type inequality. J. Approx. Theory 114, 84–97 (2002)

Konyagin, S.: Estimates for derivatives of polynomials. Dokl. Akad. Nauk SSSR 243, 1116–1118 (1978). (Russian)

Kroó, A.: On the exact constant in the $$L_2$$ L 2 Markov inequality. J. Approx. Theory 151, 208–211 (2008)

Markov, A.A.: On a question of D.I. Mendeleev. Zapiski Petersb. Akad. Nauk 62, 1–24 (1889) (in Russian). Available also at: http://www.math.technion.ac.il/hat/fpapers/mar1.pdf

Markov, V.A.: On functions which deviate least from zero in a given interval, Saint-Petersburg University (1892) (in Russian)

German translation: Math. Ann. 77, 213-258 (1916). Available also at: http://www.math.technion.ac.il/hat/fpapers/vmar.pdf

Milovanović, G.V., Mitrinović, D.S., Rassias, ThM: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

Nikolov, G.: Markov-type inequalities in the $$L_2$$ L 2 -norms induced by the Tchebycheff weights. Arch. Inequal. Appl. 1(3–4), 361–375 (2003)

Nikolov, G., Shadrin, A.: On the $$L_2$$ L 2 Markov inequality with Laguerre weight. In: Govil, N.K., et al. (eds.) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol. 117, pp. 1–17. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-49242-1_1

Nikolov, G., Shadrin, A.: Markov $$L_2$$ L 2 –inequality with the Laguerre weight. arXiv:1705.03824v1 [math.CA]

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

Schmidt, E.: Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum. Math. Ann. 119, 165–204 (1944). (in German)

Shadrin, A.: Twelve proofs of the Markov inequality. In: Dimitrov, D.K., Nikolov, G., Uluchev, R. (eds.), Approximation Theory: A volume dedicated to Borislav Bojanov, pp. 233–298. Professor Marin Drinov Academic Publishing House, Sofia (2004). Available also at: http://www.damtp.cam.ac.uk/user/na/people/Alexei/papers/markov.pdf

Turán, P.: Remark on a theorem of Ehrhard Schmidt. Mathematica (Cluj) 2, 373–378 (1960)