Nikol'skii type inequality; Bessel weight; extremal functions; extremal constants
type: Item
instance of: scholarly article
title: Nikol'skii type inequality; Bessel weight; extremal functions; extremal constants
zbMATH Open document ID: 1424.47026
published in: Analysis Mathematica
publication date: 2018-04-27T00:00:00Z
review text: For $\alpha>-1$ and $1\le q<\infty$, denote by $L_{\alpha}^q$ the space of complex valued measurable functions defined on $[0,\infty)$ for which $\Vert f\Vert_{q,\alpha}:=\left(\int_0^{\infty}\vert f(x)\vert x^{2\alpha+1}dx\right)^{\frac1q}<\infty$. For $\sigma>0$, denote by $\mathcal{E}(\sigma,q,\alpha)$, the set of even entire functions of exponential type at most $\sigma$ which belong to $L_{\alpha}^q$. Consider also the sup norm $\Vert f\Vert_C:=\sup\{\vert f(x)\vert ,\;x\in[0,\infty)\}$. The paper deals with the study of the optimal constants and the corresponding extremal functions which can appear in the following two inequalities:\begin{itemize} \item[(a)] $\Vert f\Vert_C\le M \cdot \Vert f\Vert_{q,\alpha}$ and \item[(b)] $\vert f(0)\vert \le D \cdot \Vert f\Vert_{q,\alpha}$, when $f\in \mathcal{E}(\sigma,q,\alpha)$. \end{itemize} The main result says that, for $\alpha\ge-\frac12$ and $1\le q<\infty$, we have $M(\alpha,q,\sigma)=D(\alpha,q,\sigma)$, where $M(\alpha,q,\sigma)$ and $D(\alpha,q,\sigma)$ are the optimal constants in the inequalities (a) and (b), respectively. In addition, the sets of corresponding extremal functions in (a) and (b) coincide and there is given a characterization of the extremal functions. For $1<q<\infty$, the extremal function is unique (excepting a multiplicative factor). Moreover, the uniform norm of extremal functions for inequality (a) is attained only at the point $x=0$.
reviewed by: Radu Paltanea
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 6865435
zbMATH Keywords: entire function of exponential type
zbMATH Keywords: Nikol'skii-type inequality
zbMATH Keywords: Bessel generalized translation
Wikidata QID: Q115605372
author: Vitalii Arestov
author: A. G. Babenko
author: Marina Valer'evna Deikalova
author: Ágota P. Horváth
MaRDI profile type: Publication
full work available at URL: s10476-018-0103-6
OpenAlex ID: W2802290200
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DOI: 10.1007/S10476-018-0103-6
description: scientific article; zbMATH DE number 6865435