On Constants in the Bernstein--Szeg\H{o} Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm

A. O. Leont'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 130-139

Résumé

The Weyl derivative (fractional derivative) $f_n^{(\alpha)}$ of real nonnegative order $\alpha$ is considered on the set $\mathscr{T}_n$ of trigonometric polynomials $f_n$ of order $n$ with complex coefficients. The constant in the Bernstein–Szegő inequality $\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta\|\le B_n(\alpha,\theta)\|f_n\|$ in the uniform norm is studied. This inequality has been well studied for $\alpha\ge 1$: G. T. Sokolov proved in 1935 that it holds with the constant $n^\alpha$ for all $\theta\in\mathbb{R}$. For $0\alpha1$, there is much less information about $B_n(\alpha,\theta)$. In this paper, for $0\alpha1$ and $\theta\in\mathbb{R}$, we establish the limit relation $\lim_{n\to\infty}B_n(\alpha,\theta)/n^\alpha=\mathcal{B}(\alpha,\theta)$, where $\mathcal{B}(\alpha,\theta)$ is the sharp constant in the similar inequality for entire functions of exponential type at most $1$ that are bounded on the real line. The value $\theta=-\pi\alpha/2$ corresponds to the Riesz derivative, which is an important particular case of the Weyl–Szegő operator. In this case, we derive exact asymptotics for the quantity $B_n(\alpha)=B_n(\alpha,-\pi\alpha/2)$ as $n\to\infty$

Détail

Keywords: trigonometric polynomials, entire functions of exponential type, Weyl–Szegő operator, Riesz derivative, Bernstein inequality, uniform norm.