Inequalities of the form $\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta\|_p\le B_n(\alpha,\theta)_p \|f_n\|_p$ for classical derivatives of order $\alpha\in\mathbb{N}$ and Weyl derivatives of real order $\alpha\ge 0$ of trigonometric polynomials $f_n$ of order $n\ge 1$ and their conjugates for real $\theta$ and $0\le p\le \infty$ are called Bernstein-Szego inequalities. They are generalizations of the classical Bernstein inequality ($\alpha=1$, $\theta=0$, $p=\infty$). Such inequalities have been studied for more than 90 years. The problem of studying the Bernstein-Szego inequality consists in analyzing the properties of the best (the smallest) constant $B_n(\alpha,\theta)_p$, its exact value, and extremal polynomials for which this inequality turns into an equality. G. Szego (1928), A. Zygmund (1933), and A.I. Kozko (1998) showed that, in the case $p\ge 1$ for real $\alpha\ge 1$ and any real $\theta$, the best constant $B_n(\alpha,\theta)_p$ is~$n^\alpha$. For $p=0$, Bernstein-Szego inequalities are of interest at least because the constant $B_n(\alpha,\theta)_p$ is the largest for $p=0$ over $0\le p\le\infty$. In 1981, V.V. Arestov proved that, for $r\in\mathbb{N}$ and $\theta=0$, the Bernstein inequality is true with the constant $n^r$ in the spaces $L_p$, $0\le p<1$; i.e., $B_n(r,0)_p=n^r$. In 1994, he proved that, for $p=0$ and the derivative of the conjugate polynomial of order $r\in\mathbb{N}\cup\{0 \}$, i.e., for $\theta=\pi/2$, the exact constant grows exponentially in $n$; more precisely, $B_n(r,\pi/2)_0=4^{n+o(n)}$. In two recent papers of the author (2018), a similar result was obtained for Weyl derivatives of positive noninteger order for any real $\theta$. In the present paper, we prove that the formula $B_n(\alpha,\theta)_0=4^{n+o(n)}$ holds for derivatives of nonnegative integer orders $\alpha$ and any real $\theta\neq \pi k,\,k\in\mathbb{Z}$.
Keywords: trigonometric polynomial, conjugate polynomial, Weyl derivative, Bernstein-Szego}inequality, space $L_0$
Received August 6, 2019
Revised October 21, 2019
Accepted October 28, 2019
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, 620083 Russia; N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: sinusoida2012@yandex.ru
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Cite this article as: A.O.Leont’eva. Bernstein–Szego inequality for trigonometric polynomials in the space $L_0$, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 129–135.
