Nikolskii - Bernstein constants for nonnegative entire functions of exponential type on the axis

D. V. Gorbachev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 92-103

Résumé

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 operator $B_{\alpha}=\displaystyle\frac{d^{2}}{dx^{2}}+\displaystyle\frac{2\alpha+1}{x}\,\displaystyle\frac{d}{dx}$. 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 Horváth'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 $B_{\alpha}$.

Détail

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