Asymptotics of the best constant in a certain Markov-type inequality
type: Item
instance of: scholarly article
title: Asymptotics of the best constant in a certain Markov-type inequality
zbMATH Open document ID: 1120.41015
published in: Journal of Approximation Theory
publication date: 2002-07-04T00:00:00Z
review text: Let \(\gamma_n(\alpha)\) be the best possible constant in the Markov-type inequality: \(| | p^\prime | | \leq c | | p | | , p \in \Pi_n \)( the space of complex polynomials of degree not exceeding \(n\)), where \(| | . | | \) be the weighted \(L^2\)-norm with Laguerre weight : \(w(t) = t^\alpha e^{-t}, \alpha > -1.\) \textit{P. Turán} [Mathematica, Cluj 2(25), 373--378 (1960; Zbl 0101.04702)] found the exact value of \(\gamma_{n(0)}\) which in particular implied that (*): \( lim \frac{\gamma_{n(0)}}{n} = \frac{2}{\pi}\) as \(n\) tends to infinity. However for arbitrary \(\alpha > -1\), very little is known [see P. Dörfler, Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 200, No. 1--10, 13--20 (1991; Zbl 0748.41009)]. The author obtains here the value of \( lim \frac{\gamma_n(\alpha)}{ n}\) as \(n\) tends to infinity, in terms of the first positive zero of a certain Bessel function of the first kind. In particular when \(\alpha = 0.\) this result gives the above mentioned result (*). It is also oberved that in general it is very difficult to obtain the value of \(\gamma_n(\alpha)\).
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 1763610
zbMATH Keywords: Markov inequality
zbMATH Keywords: \(L^2\)-norm
zbMATH Keywords: Laguerre weight
zbMATH Keywords: best constant
zbMATH Keywords: asymptotics
author: Peter Dörfler
reviewed by: H. P. Dikshit
MaRDI profile type: Publication
full work available at URL: jath.2001.3638
OpenAlex ID: W2083462339
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DOI: 10.1006/JATH.2001.3638
description: scientific article; zbMATH DE number 1763610