D.V. Gorbachev. Nikolskii – Bernstein constants for nonnegative entire functions of exponential type on the axis ... P. 92-103

We investigate a weighted version of the Nikolskii-Bernstein inequality
\[ \|\Lambda_{\alpha}^{k}f\|_{q,\alpha}\le \mathcal{L}(\alpha,p,q,k)\sigma^{(2\alpha+2)(1/p-1/q)+k}\|f\|_{p,\alpha},\quad \alpha\ge -1/2, \]
on the subspace $\mathcal{E}^{\sigma}\cap L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$ of entire functions of exponential type. Here $\Lambda_{\alpha}$ is the Dunkl differential-difference operator whose second power generates  the Bessel differential-difference operator $B_{\alpha}$. For $(p,q)=(1,\infty)$, we compute the following sharp constants for nonnegative functions:
\[ \mathcal{L}_{0}^{*}(\alpha)_{+}=\frac{1}{2^{2\alpha+2}},\quad \mathcal{L}_{1}^{*}(\alpha)_{+}=\frac{1}{2^{2\alpha+4}(\alpha+2)}, \]
where $\mathcal{L}_{r}^{*}(\alpha)_{+}= (\alpha+1)c_{\alpha}^{-2}\mathcal{L}(\alpha,1,\infty,2r)_{+}$ denotes the normalized Nikolskii-Bernstein constant. There are unique (up to a constant factor) extremizers $j_{\alpha+1}^{2}(x/2)$ and $x^{2}j_{\alpha+2}^{2}(x/2)$, respectively. These results are proved with the use of the Markov quadrature formula with nodes at zeros of the Bessel function and the following generalization of Arestov, Babenko, Deikalova, and Horv$\acute{\mathrm{a}}$th's recent result:
\[ \mathcal{L}(\alpha,p,\infty,2r)=\sup B_{\alpha}^{r}f(0),\quad r\in \mathbb{Z}_{+}, \]
where the supremum is taken over all even real functions on $\mathbb{R}$ belonging to $\mathcal{E}_{p,\alpha}^{1}$. Our approach is based on the one-dimensional Dunkl harmonic analysis. In particular, we use the even positive Dunkl-type generalized translation operator $T_{\alpha}^{t}$, which is bounded on $L^{p}(\mathbb{R},|t|^{2\alpha+1}\,dt)$ with constant 1, is invariant on the subspace $\mathcal{E}_{p,\alpha}^{\sigma}$, and commutes with $B_{\alpha}$.

Keywords: weighted Nikolskii-Bernstein inequality, sharp constant, entire function of exponential type, Dunkl transform, generalized translation operator, Bessel function

Received September 05, 2018

Revised November 15, 2018

Accepted November 19, 2018

Funding Agency: This work was supported by the Russian Science Foundation (project no. 18-11-00199).

Dmitry Viktorovich Gorbachev, Dr. Phys.-Math. Sci., Prof., Tula State University, Tula, 300012 Russia, e-mail: dvgmail@mail.ru

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