Bernstein–Szegő Inequality for the Weyl Derivative of Trigonometric Polynomials in $L_0$

A. O. Leont'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 199-207

Résumé

In the set $\mathscr{T}_n$ of trigonometric polynomials $f_n$ of order $n$ with complex coefficients, we consider Weyl (fractional) derivatives $f_n^{(\alpha)}$ of real nonnegative order $\alpha$. The inequality $\left\|D^\alpha_\theta f_n\right\|_p\le B_n(\alpha,\theta)_p \|f_n\|_p$ for the Weyl–Szegő operator $D^\alpha_\theta f_n(t)=f_n^{(\alpha)}(t)\cos\theta+\tilde{f}_n^{(\alpha)}(t)\sin\theta$ in the set $\mathscr{T}_n$ of trigonometric polynomials is a generalization of Bernstein's inequality. Such inequalities have been studied for 90 years. G. Szegő obtained the exact inequality $\left\|f_n'\cos\theta+\tilde{f}_n'\sin\theta\right\|_\infty \leq n\left\|f_n\right\|_\infty$ in 1928. Later, A. Zygmund (1933) and A.I. Kozko (1998) showed that, for $p\ge 1$ and real $\alpha\ge 1$, the constant $B_n(\alpha,\theta)_p$ equals $n^\alpha$ for all $\theta\in\mathbb{R}$. The case $p=0$ is of additional interest because it is in this case that $B_n(\alpha,\theta)_p$ is largest over $p\in[0,\infty]$. In 1994 V. V. Arestov showed that, for $\theta=\pi/2$ (in the case of the conjugate polynomial) and integer nonnegative $\alpha$, the quantity $B_n(\alpha,\pi/2)_0$ grows exponentially in $n$ as $4^{n+o(n)}$. It follows from his result that the behavior of the constant for $\theta\neq 2\pi k$ is the same. However, in the case $\theta=2\pi k$ and $\alpha\in\mathbb{N}$, Arestov showed in 1979 that the exact constant is $n^\alpha$. The author investigated Bernstein's inequality in the case $p=0$ for positive noninteger $\alpha$ earlier (2018). The logarithmic asymptotics of the exact constant was obtained: $\sqrt[n]{B_n(\alpha,0)_0}\to 4$ as $n\to\infty$. In the present paper, this result is generalized to all $\theta \in \mathbb{R}$.

Détail

Keywords: trigonometric polynomial, Weyl derivative, Bernstein–Szeg\H{o
Mots-clés : conjugate polynomial

Bibliographie

[1] Weyl H., “Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung”, Vierteljahrcsschrift der Naturforschenden Gesellschaft in Zurich, 62:1–2 (1917), 296–302 | MR | Zbl

[2] Samko S.G., Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 638 pp.

[3] Arestov V.V., Glazyrina P.Yu., “Bernstein-Szego inequality for fractional derivatives of trigonometric polynomials”, Proc. Steklov Inst. Math., 288, Suppl. 1, 2015, 13–28 | DOI | MR

[4] Szegö G., “Über einen Satz des Herrn Serge Bernstein”, Schrift. Königsberg. Gelehrten Gesellschaft, 5:4 (1928), 59–70 | Zbl

[5] Zigmund A., Trigonometricheskie ryady, v 2 t., v. 1, Mir, M., 1965, 616 pp. ; т. 2, 538 с. | MR

[6] Kozko A.I., “The exact constants in the Bernstein-Zygmund-Szegö inequalities with fractional derivatives and the Jackson-Nikolskii inequality for trigonometric polynomials”, East J. Approx, 4:3 (1998), 391–416 | MR | Zbl

[7] Arestov V.V., “Neravenstvo Sege dlya proizvodnykh sopryazhennogo trigonometricheskogo polinoma v $L_0$”, Mat. zametki, 56:6 (1994), 10–26 | Zbl

[8] Arestov V.V., “O neravenstvakh S. N. Bernshteina dlya algebraicheskikh i trigonometricheskikh polinomov”, Dokl. AN SSSR, 246:6 (1979), 1289–1292 | MR | Zbl

[9] Arestov V.V., “Ob integralnykh neravenstvakh dlya trigonometricheskikh polinomov i ikh proizvodnykh”, Izv. AN SSSR. Ser. matematicheskaya, 45:1 (1981), 3–22

[10] Adamov A.N., “O konstante v neravenstve Sege dlya proizvodnykh sopryazhennykh trigonometricheskikh polinomov v $L_0$”, Vestnik Od. nats. un-ta. Matematiki i mekhanika, 19:1(21) (2014), 7–15

[11] Leonteva A.O., “Neravenstvo Bernshteina dlya proizvodnykh Veilya trigonometricheskikh polinomov v prostranstve $L_0$”, Mat. zametki, 104:2 (2018), 255–264 | DOI | MR | Zbl

[12] Arestov V.V., “Integralnye neravenstva dlya algebraicheskikh mnogochlenov na edinichnoi okruzhnosti”, Mat. zametki, 48:4 (1990), 7–18

[13] Polia G., Sege G., Zadachi i teoremy iz analiza, v 2 t., v. 1, Nauka, M., 1978, 391 pp.

[14] Popov N.V., “O neravenstve S. N. Bernshteina”, Sovremennye problemy teorii funktsii i ikh prilozheniya, materialy 19 Mezhdunar. Saratovskoi zimnei shk., posvyaschennoi 90-letiyu so dnya rozhdeniya akad. P. L. Ulyanova, OOO Izd-vo “Nauchnaya kniga”, Saratov, 2018, 380 pp.