Bernstein--Szeg\H{o} 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