Bounds of the Nikol'skii Polynomial Constants in $L^{p}$ with Gegenbauer Weight

D. V. Gorbachev ; I. A. Martyanov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 126-137

Résumé

We study bounds and the asymptotic behavior as $n\to \infty$ of a sharp Nikol'skii constant in the inequality $\|u\|_{\infty}\le \mathcal{C}_{\alpha}(n)\|u\|_{p}$ for trigonometric and algebraic polynomials of degree at most $n$ in the space $L^{p}$ on $(-\pi,\pi]$ with the periodic Gegenbauer weight $|\!\sin x|^{2\alpha+1}$ and on $ [-1,1] $ with the algebraic Gegenbauer weight $(1-x^{2})^{\alpha}$, respectively. We prove that $\mathcal{C}_{\alpha}(n)\sim \mathcal{L}_{p}n^{(2\alpha+2)/p}$ for $p\ge 1$ and all $\alpha\ge-1/2$, where $\mathcal{L}_{p}$ is a sharp Nikol'skii constant for entire functions of exponential type at most $1$ in the space $L^{p}$ on $\mathbb{R}$ with the power weight $|x|^{2\alpha+1}$. Moreover, we give explicit bounds of the form $$ n^{(2\alpha+2)/p}\mathcal{L}_{p}\le \mathcal{C}_{\alpha}(n)\le (n+2s_{p,\alpha})^{(2\alpha+2)/p}\mathcal{L}_{p},\quad n\ge 0, $$ from which this asymptotics follows. These bounds make it possible to refine the known estimates of the Nikol'skii constants. We consider this approach using the example of the algebraic Nikol'skii constant for $\alpha=0$. Here we apply the characterization of the extremal polynomials from the works of D. Amir and Z. Ziegler and of V.V. Arestov and M.V. Deikalova. Our statements generalize the well-known results of S.B. Stechkin ($p=1$) and E. Levin and D. Lubinsky ($p>0$) in the trigonometric case for $\alpha=-1/2$ and M.I. Ganzburg in the algebraic case for $\alpha=0$. For half-integer $\alpha=d/2-1$ and $p\ge 1$, our asymptotics can be derived from the asymptotics of the multidimensional Nikol'skii constant for spherical polynomials in the space $L^{p}$ on the sphere $\mathbb{S}^{d}$ proved by F. Dai, D. Gorbachev, and S. Tikhonov. Our proof is much simpler, but it does not cover the case $p1$.

Détail

Keywords: Nikol'skii inequality, asymptotic behavior, trigonometric polynomial, entire function of exponential type, Gegenbauer weight.
Mots-clés : sharp constant, algebraic polynomial

Bibliographie

[1] Amir D., Ziegler Z., “Polynomials of extremal $L_p$-norm on the $L_\infty$-unit sphere”, J. Approx. Theory, 18 (1976), 86–98 | DOI

[2] Arestov V.V., Deikalova M.V., “Neravenstvo Nikolskogo dlya algebraicheskikh mnogochlenov na mnogomernoi evklidovoi sfere”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:2 (2013), 34–47

[3] 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

[4] Arestov V., Babenko A., Deikalova M., Horvath A., “Nikol'skii inequality between the uniform norm and integral norm with Bessel weight for entire functions of exponential type on the half-Line”, Anal. Math., 44:1 (2018), 21–42 | DOI

[5] Dai F., Gorbachev D., Tikhonov S., “Estimates of the asymptotic Nikolskii constants for spherical polynomials”, [e-resource], 27 pp., arXiv: https://arxiv.org/pdf/1907.03832

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

[7] Dai F., Xu Y., Approximation theory and harmonic analysis on spheres and balls, Springer-Verlag, N Y, 2013, 440 pp. | DOI

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

[9] Ganzburg M.I., “Sharp constants of approximation theory. I. Multivariate Bernstein-Nikolskii type inequalities”, J. Fourier Anal. Appl., 26:11 (2020) | DOI

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

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

[12] Gorbachev D.V., “Integralnaya zadacha Konyagina i $(C,L)$-konstanty Nikolskogo”, Tr. In-ta matematiki i mekhaniki UrO RAN, 11:2 (2005), 72–91

[13] 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

[14] Gorbachev D.V., Ivanov V.I., Tikhonov S.Yu., “Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications”, Constr. Approx., 49:3 (2019), 555–605 | DOI

[15] Gorbachev D.V., Martyanov I.A., “O vzaimosvyazi konstant Nikolskogo dlya trigonometricheskikh polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 19:2 (2018), 80–89 | DOI

[16] Ibragimov I.I., “Ekstremalnye zadachi v klasse trigonometricheskikh polinomov”, Dokl. AN SSSR, 121:3 (1958), 415–417

[17] Kamzolov A.I., “O priblizhenii funktsii na sfere $S^{n}$”, Serdika, 84:1 (1984), 3–10

[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

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

[20] Milovanovic G.V., Mitrinovic D.S., Rassias Th.M., Topics in polynomials: Extremal problems, inequalities, zeros, World Scientific Publ. Co., Singapore, 1994, 836 pp.

[21] Nikolskii S.M., “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Tr. MIAN SSSR, 38 (1951), 244–278

[22] Nursultanov E., Ruzhansky M., Tikhonov S., “Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds”, Ann. Sc. Norm. Super. Pisa Cl. Sci., XVI:3 (2016), 981–1017 | DOI

[23] Pesenson I., “Bernstein-Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds”, J. Approx. Theory, 150:2 (2008), 175–198 | DOI

[24] Taikov L.V., “Odin krug ekstremalnykh zadach dlya trigonometricheskikh polinomov”, Uspekhi mat. nauk, 20:3 (1965), 205–211