A new characterization of Hermite polynomials
type: Item
instance of: scholarly article
title: A new characterization of Hermite polynomials
zbMATH Open document ID: 0612.33008
DOI: 10.1007/BF01956321
author: A. K. Varma
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: Consider the polynomials on \((-\infty,\infty)\) equipped with the norm derived from the integral inner product with weight function \(\exp (-x^ 2):\) \[ \| P_ n\|^ 2=\int^{\infty}_{-\infty}P^ 2_ n(t)\exp (-t^ 2)dt \] \((P_ n:\) a general n-th degree polynomial). Based on the inequality \(\| P_ n'\| \leq a_ n\| P_ n\|^ 2+b_ n\| P_ n''\|^ 2\) with \(a_ n=(2n-1)\), \(b_ n=2n^ 2/(2n-1)\), which turns into an equality iff \(P_ n\) is - up to a multiplicative constant - the Hermite polynomial of degree n w.r.t. the weight \(\exp (-x^ 2)\) on \((-\infty,\infty)\), the author gives bounds for \(\| P_ n^{(q)}\|^ 2\) by a linear combination (coefficients depending on n) of \(\| P_ n\|^ 2\) and \(\| P_ n^{(r)}\|^ 2\) for \(1\leq q\leq r-1\) and \(r=2,3,4\). All inequalities characterize the Hermite polynomials: those are the only ones (up to a multiplicative constant) which lead to equality.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 3989787
zbMATH Keywords: Landau inequalities
zbMATH Keywords: Hermite polynomial
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
cites work: Q5669818
cites work: The Elementary Cases of Landau's Problem of Inequalities Between Derivatives
cites work: Q5637340
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description: scientific article; zbMATH DE number 3989787