Markov--Bernstein--Nikol'skii constants for polynomials in~$L^{p}$-space with the Gegenbauer weight

D. V. Gorbachev ; I. A. Martyanov

Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 29-44.

Résumé

We study the sharp Markov–Bernstein–Nikol'skii inequality of the form $\|D^{s}u\|_{\infty} \le C_{p}(n;s)\times\\\times\|u\|_{p}$, $p\in [1,\infty]$ for trigonometric and algebraic polynomials $u$ of degree at most $n$ in the weighted space $L^{p}$ with the Gegenbauer–Dunkl differential operator $D$. In particular cases, these inequalities are reduced to the classical inequalities of approximation theory of the Markov, Bernstein, and Nikol'skii type, to which numerous papers are devoted. We apply the results of V.A. Ivanov (1983, 1992), V.V. Arestov and M.V. Deikalova (2013, 2015), F. Dai, D.V. Gorbachev and S.Yu. Tikhonov (2020) for algebraic constants in $L^{p}$ on compact Riemannian manifolds of rank 1 (including the Euclidean sphere) and an interval with Gegenbauer weight, refer to the works of E. Levin and D. Lubinsky (2015), M.I. Ganzburg (2017, 2020), a review of the classic results of G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias (1994). Earlier we studied the case $s=0$. In this paper, we consider the case $s\ge 0$. Our main result is to prove the existence in the trigonometric case for even $s=2r$ of extremal polynomials $u_{*}$ that are real, even, and $C(n;s)=\frac{|D^{s}u_{*}(0)|}{\|u_{*}\|_{p}}$. With the help of this fact, the relationship with the algebraic constant for the Gegenbauer weight is proved. On the one hand, this relationship allows to automatically characterize extremal algebraic polynomials. On the other hand, well-known algebraic results carry over to a more general trigonometric version. The main method of proof is the application of the Gegenbauer–Dunkl harmonic analysis constructed by D.V. Chertova (2009). As a consequence, we give the explicit constants for $p=2, \infty$ (using the results of V.A. Ivanov), we give the relations of orthogonality and duality (proved by methods of convex analysis from approximation theory), we establish one asymptotic result of the Levin–Lubinsky type (due to the connection with the multidimensional Nikol'skii constant for spherical polynomials).

Détail

Keywords: trigonometric polynomial, algebraic polynomial, the Bernstein–Nikolskii constant, the Markov–Nikolskii constant, the Gegenbauer weight.

Bibliographie

[1] Arestov V. V., Deikalova M. V., “Neravenstvo Nikolskogo dlya algebraicheskikh mnogochlenov na mnogomernoi evklidovoi sfere”, Tr. IMM UrO RAN, 19, no. 2, 2013, 34–47

[2] Arestov V. V., Glazyrina P. Yu., “Neravenstvo Bernshteina–Sege dlya drobnykh proizvodnykh trigonometricheskikh polinomov”, Tr. IMM UrO RAN, 20, no. 1, 2014, 17–31

[3] Gorbachev D. V., Dobrovolskii N. N., “Konstanty Nikolskogo v prostranstvakh $L^{p}(\mathbb{R},|x|^{2\alpha+1}~dx)$”, Chebyshevskii sb., 19:2 (2018), 67–79 | DOI | MR | Zbl

[4] Gorbachev D. V., Martyanov I. A., “Vzaimosvyaz mezhdu konstantami Nikolskogo–Bernshteina dlya trigonometricheskikh polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sbornik, 20:3 (2019), 143–153 | DOI | MR | Zbl

[5] Ivanov V. A., “O neravenstvakh Bernshteina–Nikolskogo i Favara na kompaktnykh odnorodnykh prostranstvakh ranga 1”, UMN, 38:3(231) (1983), 179–180 | MR | Zbl

[6] Ivanov V. A., “Tochnye rezultaty v zadache o neravenstve Bernshteina–Nikolskogo na kompaktnykh simmetricheskikh rimanovykh prostranstvakh ranga 1”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Ch. 14, Tr. MIAN SSSR, 194, Nauka, M., 1992, 111–119

[7] Ivanov V. I., “Nekotorye ekstremalnye svoistva polinomov i obratnye neravenstva teorii priblizheniya”, Priblizhenie funktsii polinomami i splainami, Tr. MIAN SSSR, 145, 1980, 79–110 | Zbl

[8] Konyagin S. V., “Otsenki proizvodnykh ot mnogochlenov”, Dokl. AN SSSR, 243:5 (1978), 1116–1118 | MR | Zbl

[9] Martyanov I. A., “Konstanta Nikolskogo dlya trigonometricheskikh polinomov s periodicheskim vesom Gegenbauera”, Chebyshevskii sbornik, 21:1 (2020), 247–258 | DOI | MR

[10] Sege G., Ortogonalnye mnogochleny, Fizmatlit, M., 1962

[11] Chertova D. V., “Teoremy Dzheksona v prostranstvakh $L_{p}$, $1\leq p2$, s periodicheskim vesom Yakobi”, Izv. TulGU. Estestvennye nauki, 2009, no. 1, 5–27 | MR

[12] Arestov V., Deikalova M., “Nikol'skii inequality between the uniform norm and $L_{q}$-norm with ultraspherical weight of algebraic polynomials on an interval”, Comput. Methods Funct. Theory, 15:4 (2015), 689–708 | DOI | MR | Zbl

[13] Dai F., Gorbachev D., Tikhonov S., Estimates of the asymptotic Nikolskii constants for spherical polynomials, arXiv: 1907.03832 | MR

[14] Dai F., Gorbachev D., Tikhonov S., “Nikolskii constants for polynomials on the unit sphere”, J. d'Anal. Math., 140:1 (2020), 161–185 | DOI | MR | Zbl

[15] Ganzburg M. I., “Sharp constants in V.A. Markov–Bernstein type inequalities of different metrics”, J. Approx. Theory, 215 (2017), 92–105 | DOI | MR | Zbl

[16] Ganzburg M. I., “Sharp constants of approximation theory. III. Certain polynomial inequalities of different metrics on convex sets”, J. Approx. Theory, 252 (2020) | DOI | MR | Zbl

[17] Ganzburg M. I., Tikhonov S. Y., “On sharp constants in Bernstein–Nikolskii inequalities”, Constr. Approx., 45:3 (2017), 449–466 | DOI | MR | Zbl

[18] Levin E., Lubinsky D., “Asymptotic behavior of Nikolskii constants for polynomials on the unit circle”, Comput. Methods Funct. Theory, 15:3 (2015), 459–468 | DOI | MR | Zbl

[19] Milovanović G.V., Mitrinović D.S., Rassias Th.M., Topics in polynomials: Extremal problems, inequalities, zeros, World Scientific Publ. Co, Singapore, 1994 | MR | Zbl

[20] Shapiro H., Topics in approximation theory, Lecture Notes in Mathematics, 187, Springer-Verlag, Berlin–Heidelberg, 1971 | DOI | MR | Zbl