Bernstein--Szeg{\H o} inequality for trigonometric polynomials in the space $L_0$ with a constant greater than classical

A. O. Leont'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 128-136

Résumé

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 $\alpha2n-2$ remans open. In the present paper, we prove that for any $0\le\alpha$ one can find $\theta^*(\alpha)$ such that $B_n(\alpha, \theta^*(\alpha))_0>n^\alpha$.

Détail

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