A.O. Leont’eva. On constants in the Bernstein–Szegö inequality for the Weyl derivative of order less than unity of trigonometric polynomials and entire functions of exponential type in the uniform norm ... P. 130-139

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. We study the question about the constant in the Bernstein—Szegö inequality $\Bigl\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta\Bigr\|\le B_n(\alpha,\theta)\|f_n\|$ in the uniform norm. 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—Szegö operator. In this case, we derive an exact asymptotic expansion for the quantity $B_n(\alpha)=B_n(\alpha,-\pi\alpha/2)$ as $n\to\infty$.

Keywords: trigonometric polynomials, entire functions of exponential type, Weyl—Szegö operator, Riesz derivative, Bernstein inequality, uniform norm

Received July 3, 2023

Revised August 8, 2023

Accepted August 14, 2023

Funding Agency: This work was performed as a part of the research conducted in the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. 075-02-2023-913).

Anastasiya Olegovna Leont’eva, Cand. Sci. (Phys.-Math.), 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. On constants in the Bernstein–Szegö inequality for the Weyl derivative of order less than unity of trigonometric polynomials and entire functions of exponential type in the uniform norm. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 130–139; Proceedings of the Steklov Institute of Mathematics (Suppl)., 2023, Vol. 323, Suppl. 1, pp. S146–S154.