An extremal property of Hermite polynomials.
type: Item
instance of: scholarly article
title: An extremal property of Hermite polynomials.
zbMATH Open document ID: 1042.41007
author: G. Nikolov
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: Consider the sequence of the Hermite polynomials \(H_m(x)=(-1)^m e^{x^2}\frac{d^m}{dx^m} \{e^{-x^2}\}\), \((m=0,1,\dots)\), i.e. orthogonal polynomials in \(L_2(\mathbb R,w)\), where \(w(x)=\exp(-x^2)\). The main result of the paper is the following Duffin-Schaeffer type inequality. If \(f\) is a polynomial on \(\mathbb R\) of degree at most \(n\) such that \(| f| \leq| H_n| \) at the zeros of \(H_{n+1}\), then for \(k=1,\dots,n\), \(\| f^{(k)}\| \leq\| H_n^{(k)}\| .\) Moreover, equality in the above inequality holds if and only if \(f=cH_n\) with \(| c| =1\).
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 2056374
zbMATH Keywords: Gauss-type quadrature formulae
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Duffin-Schaeffer type inequality
reviewed by: Wiesław Pleśniak
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.002
OpenAlex ID: W2000026194
cites work: Q4885025
cites work: Markov interlacing property for perfect splines
cites work: Duffin and Schaeffer type inequality for ultraspherical polynomials
cites work: Markov inequalities for weight functions of Chebyshev type
cites work: On a conjecture concerning monotonocity of zeros of ultraspherical polynomials
cites work: A Refinement of an Inequality of the Brothers Markoff
cites work: On the inequalities of Bernstein-Markoff for an interval
cites work: Quadrature formulae and polynomial inequalities
cites work: Q4938059
cites work: On the inequality of I. Schur
cites work: On certain Duffin and Schaeffer type inequalities
cites work: An inequality for polynomials with elliptic majorant
cites work: Inequalities of Duffin-Schaeffer type
cites work: Markov-Duffin-Schaeffer Inequality for Polynomials With a Circular Majorant
cites work: Polynomials with a parabolic majorant and the Duffin-Schaeffer inequality
cites work: Some new characterizations of the Chebyshev polynomials
cites work: Q4022921
cites work: On some extremal properties of algebraic polynomials
cites work: On mechanical quadratures formulae involving the classical orthogonal polynomials
DOI: 10.1016/J.JMAA.2003.10.002
Recommended article: Some properties of the Hermite polynomials
Recommended article: A characterization of Hermite polynomials
Recommended article: A generalization of Hermite polynomials
Recommended article: On Hermite-Hermite matrix polynomials
Recommended article: Q5754211
Recommended article: An extremal property of Chebyshev polynomials
Recommended article: Q5369760
description: scientific article; zbMATH DE number 2056374