Generalized Christoffel functions for Jacobi-exponential weights on \([-1, 1]\)
type: Item
instance of: scholarly article
title: Generalized Christoffel functions for Jacobi-exponential weights on \([-1, 1]\)
zbMATH Open document ID: 1374.42050
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Acta Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
review text: The authors study the asymptotic behavior of the \(L^p\)-Christoffel functions \[ \lambda_{p,n}(w;z):=\inf\,\frac{\| Pw\|_{L^p}^p}{| P(z)| ^p}\,, \qquad 0<p<\infty, \] where infimum is taken over all algebraic polynomials \(P\) of degree at most \(n\). The weights \(w\) on \([-1,1]\) are of the form \[ w(x)=e^{-Q(x)}\,\prod_{i=1}^r | x-t_i| ^{p_i}, \qquad t_i\in [-1,1], \quad p_i>\frac1p\,, \quad i=1,2,\ldots,r, \] and \(Q\) belongs to the Lubinsky-Levin class of functions. The main result states that \[ \lambda_{p,n}(w;x)\sim \frac{\sqrt{1-x^2}}n\,e^{-pQ(x)}\,\prod_{i=1}^r \Bigl(| x-t_i| +\frac1n\Bigr)^{p_i} \] uniformly on a certain subset of \([-1,1]\).
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 6699156
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: generalized Christoffel function
zbMATH Keywords: Jacobi-exponential weight
zbMATH Keywords: restricted range inequality
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: s10474-015-0542-5
OpenAlex ID: W4253078951
cites work: Q5812325
cites work: Orthonormal polynomials with generalized Freud-type weights.
cites work: Christoffel functions and orthogonal polynomials for exponential weights on [-1,1]
cites work: Orthogonal polynomials for exponential weights
cites work: Orthogonal polynomials for exponential weights \(x^{2\rho} e^{-2Q(x)}\) on [0,\(d\))
cites work: Orthogonal polynomials for exponential weights \(x^{2\rho} e^{-2Q(x)}\) on \([0,d)\). II.
cites work: \(L^{p}\)-convergence of Fourier sums with exponential weights on \((-1,1)\)
cites work: Orthogonal polynomials
cites work: Polynomial inequalities and embedding theorems with exponential weights on \((-1, 1)\)
cites work: Generalized Christoffel functions for Jacobi-exponential weights
DOI: 10.1007/S10474-015-0542-5
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description: scientific article; zbMATH DE number 6699156