A.O. Leont’eva. Bernstein–Szego inequality for the Weyl derivative of trigonometric polynomials in $L_0$ ... P. 199-207

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-Szego 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 the Bernstein inequality. Such inequalities have been studied for 90 years. G. Szego 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 on, A. Zygmund (1933) and A.I. Kozko (1998) showed that, for $p\ge 1$ and real $\alpha\ge 1$, the constant $B_n(\alpha,\theta)_p$ is equal to $n^\alpha$ 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 (1994) 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 the Bernstein 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 $\theta\in\mathbb{R}$.

Keywords: trigonometric polynomial, Weyl derivative, conjugate polynomial, Bernstein-Szego inequality, space $L_0$

Received July 01, 2018

Revised October 01, 2018

Accepted October 15, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Anastasia Olegovna Leont’eva, doctoral student, Ural Federal University, Yekaterinburg, 620002 Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: sinusoida2012@yandex.ru

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