scholarly journals Asymptotics for orthogonal polynomials and Christoffel functions on a ball

1996 ◽  
Vol 3 (2) ◽  
pp. 257-272 ◽  
Author(s):  
Yuan Xu
Keyword(s):  
Orthogonal Polynomials ◽  
Christoffel Functions

Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials

2013 ◽  
Vol 65 (3) ◽  
pp. 600-620 ◽  
Author(s):  
A. Kroó ◽  
D. S. Lubinsky
Keyword(s):  
Orthogonal Polynomials ◽  
Compact Set ◽  
Christoffel Functions ◽  
Analytic Parametrization ◽  
Multivariate Orthogonal Polynomials ◽  
General Conditions

AbstractWe establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain. In particular, this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

scholarly journals Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)

2020 ◽  
Vol 18 (1) ◽  
pp. 138-149
Author(s):  
Rong Liu
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Restricted Range ◽  
Exponential Weights ◽  
Christoffel Functions

Abstract Let Wα,ρ = xα(1 – x2)ρe–Q(x), where α > – $\begin{array}{} \displaystyle \frac12 \end{array}$ and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights $\begin{array}{} \displaystyle W^2_{\alpha, \rho} \end{array}$ and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.

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