On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm

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 $0<\alpha<1$, there is much less information about $B_n(\alpha,\theta)$. In this paper, for $0<\alpha<1$ 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 $n\to\infty$