A.O. Leont’eva. Bernstein–Szegő inequality for trigonometric polynomials in the space $L_0$ with a constant greater than classical... P. 128-136

In the set $\mathscr{T}_n$ of trigonometric polynomials $f_n$ of order $n$ with complex coefficients, the Weyl derivative (fractional derivative) $f_n^{(\alpha)}$ of real nonnegative order $\alpha$ is considered. The exact constant $B_n(\alpha,\theta)_p$ in Bernstein—Szegő inequality $\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta \|_p\le B_n(\alpha,\theta)_p\|f_n\|_p$ is analyzed. Such inequalities have been studied for more than 90 years. It is known that, for $1\le p\le\infty$, $\alpha\ge 1$, and $\theta\in\mathbb R$, the constant takes the classical value $B_n(\alpha,\theta)_p=n^\alpha$. The case $p=0$ is of interest at least because the constant $B_n(\alpha,\theta)_0$ takes the maximum value in $p$ for $p\in[0,\infty]$. V.V. Arestov proved that, for $r\in\mathbb N$, the Bernstein inequality in $L_0$ holds with the constant $B_n(r,0)_0=n^r$, and the constant $B_n(\alpha,\pi/2)_0$ in the Szegő inequality in $L_0$ behaves as $4^{n+o(n)}$. V.V. Arestov in 1994 and V.V. Arestov and P.Yu. Glazyrina in 2014 studied the question of conditions on the parameters $n$ and $\alpha$ under which the constant in the Bernstein—Szegő inequality takes the classical value $n^\alpha$. Recently, the author has proved Arestov and Glazyrina's conjecture that the  Bernstein—Szegő inequality holds with the constant $n^\alpha$ for $\alpha\ge 2n-2$ and all $\theta\in\mathbb R$. The question about the exactness of the bound $\alpha=2n-2$, more precisely, the question of the best constant for $\alpha<2n-2$ remans open. In the present paper, we prove that for any $0\le\alpha<n$ one can find $\theta^*(\alpha)$ such that $B_n(\alpha, \theta^*(\alpha))_0>n^\alpha$.

Keywords: trigonometric polynomials, Weyl derivative, Bernstein—Szegő inequality, space $L_0$

Received May 20, 2022

Revised September 25, 2022

Accepted October 3, 2022

Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-874).

Anastasiya Olegovna Leont’eva, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: lao-imm@yandex.ru


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Cite this article as: A.O. Leont’eva. Bernstein–Szegő inequality for trigonometric polynomials in the space L0 with a constant greater than classical, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2022, vol. 28, no. 4, pp. 128–136