Bounds of the Nikol'skii polynomial constants in \(L^p\) with Gegenbauer weight
type: Item
instance of: scholarly article
title: Bounds of the Nikol'skii polynomial constants in \(L^p\) with Gegenbauer weight
zbMATH Open document ID: 1494.41005
DOI: 10.1134/S0081543821060109
author: I. A. Mart'yanov
author: Dmitry Gorbachev
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2022-01-28T00:00:00Z
review text: The authors estimate a sharp Nikol'skii constant in the inequality \(\|u\|_{\infty}\leq \mathcal{C}_{\alpha}(n)\|u\|_p\) for trigonometric and algebraic polynomials of degree at most \(n\) in the space \(L^p\) on \((-\pi,\pi]\) with the periodic Gegenbauer weight \(|\sin x|^{2\alpha +1}\) and on \([-1, 1]\) with the algebraic Gegenbauer weight \((1-x^2)^{\alpha}\), respectively. Suppose that \(p\geq 1\) and let \(\mathcal{L}_p\) be a sharp Nikol'skii constant for entire functions of exponential type at most 1 in the space \(L^p\) on \(\mathbb{R}\) with the power weight \(|x|^{2\alpha +1}\). It is proved that, for all \(\alpha\geq -1/2\), \[ n^{(2\alpha + 2)/p}\mathcal{L}_p\leq\mathcal{C}_{\alpha}(n)\leq (n + 2s_{p,\alpha})^{(2\alpha + 2)/p}\mathcal{L}_p, \] where \(s_{p,\alpha}\) does not depend on \(n\). These bounds make it possible to refine some known estimates of the Nikol'skii constants.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 7466407
zbMATH Keywords: Nikol'skii inequality
zbMATH Keywords: sharp constant
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: trigonometric polynomial
zbMATH Keywords: algebraic polynomial
zbMATH Keywords: entire function of exponential type
zbMATH Keywords: Gegenbauer weight
reviewed by: Yuri A. Farkov
MaRDI profile type: Publication
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full work available at URL: s0081543821060109
OpenAlex ID: W4205927651
description: scientific article; zbMATH DE number 7466407