Strong asymptotics for Sobolev orthogonal polynomials
type: Item
instance of: scholarly article
title: Strong asymptotics for Sobolev orthogonal polynomials
zbMATH Open document ID: 0937.42011
DOI: 10.1007/BF02791131
author: H. Pijeira Cabrera
author: Andrei Martínez Finkelshtein
published in: Journal d'Analyse Mathématique
publication date: 2000-05-15T00:00:00Z
review text: The authors prove strong asymptotic results for the polynomials \(Q_n\), orthogonal with respect to the inner product \[ \langle f,g,\rangle=\sum_{k=0}^n \int_{\Delta_k} f^{(k)}(x)g^{(k)}(x) d\mu_k(x), \] with \(\{\mu_k\}\) measures supported in \([-1,1]\), satisfying Szegő's condition. The main results concern the following situation: \(\Delta_k\subset\Delta_m\;(0\leq k\leq m-1),\;\Delta_m=[-1,1]\), \(\kappa_n=\langle Q_n,Q_n\rangle\), \(\mathcal F\) the Szegő extremal function corresponding to \(\Delta_m\), \(\varphi(z)\) the conformal map of \([-1,1]\) to the exterior of \(|z|=1/2\). \textbf{A.} If \(d\mu_m(x)=\rho(x)dx\) with \(\rho\) belonging to the Szegő class on \([-1,1]\), \(\{\mu_k\}_0^{m-1}\) finite positive Borel measures, \(\gamma_k(\rho)= \int|P_k(\rho;x)|^2d\mu_m(x)\) with \(P_k\) the orthogonal polynomial of degree \(k\) with respect to \(d\mu_m(x)\), then \[ \lim_{n\rightarrow\infty} {\kappa_n\over n^{2m}\gamma_{n-m}(\rho)}=1, \quad \lim_{n\rightarrow\infty} {Q_n^{(m)}\over n^m\varphi^{n-m}(z)}= {\mathcal F}(z), \quad \lim_{n\rightarrow\infty} {Q_n^{(m)}\over n^mP_{n-m}(\rho;z)}=1, \] the last two limits uniformly on compact subsets of \([-1,1]\). \textbf{B.} If \(\{\mu_k\}_0^m\) are finite, positive, absolutely continuous Borel measures supported on \([-1,1]\), all belonging to the Szegő class, then \[ \lim_{n\rightarrow\infty} {Q_n^{(k)}\over n^k\varphi^{n-k}(z)}= {{\mathcal F}(z)\over (\varphi'(z))^{n-k}}, \qquad \lim_{n\rightarrow\infty} {Q_n^{(k)}\over n^kP_{n-k}(d\mu_m;z)} ={1\over (\varphi'(z))^{n-k}}, \] for \(0\leq k\leq m\), uniformly on compact subsets of \([-1,1]\). An excellent contribution to the steadily growing insight into the nature of Sobolev orthogonal polynomials. The results found, do not ask for information about the location of the zeros of \(Q_n\) (which might be outside the complex hull of the supports of the underlying measures).
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1369958
zbMATH Keywords: Sobolev orthogonality
zbMATH Keywords: strong asymptotics
zbMATH Keywords: Szegő's class
MaRDI profile type: Publication
cites work: Orthogonal polynomials on Sobolev spaces: Old and new directions
cites work: The moment problem for a Sobolev inner product
cites work: Q5618011
cites work: Zeros and critical points of Sobolev orthogonal polynomials
cites work: Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product
cites work: Zero location and \(n\)th root asymptotics of Sobolev orthogonal polynomials
cites work: Relative asymptotics for orthogonal polynomials with a Sobolev inner product
cites work: Asymptotic properties of Sobolev orthogonal polynomials
cites work: Bernstein-Szegő's theorem for Sobolev orthogonal polynomials
cites work: Q4892493
cites work: Q5289656
cites work: Q4061445
cites work: Extremal polynomials associated with a system of curves in the complex plane
full work available at URL: https://doi.org/10.1007/bf02791131
OpenAlex ID: W2009982726
description: scientific article; zbMATH DE number 1369958