Menke points on the real line and their connection to classical orthogonal polynomials
type: Item
instance of: scholarly article
title: Menke points on the real line and their connection to classical orthogonal polynomials
zbMATH Open document ID: 1189.33018
author: J. Blot
published in: Journal of Computational and Applied Mathematics
publication date: 2009-12-18T00:00:00Z
review text: Let \(q\) and \(p\) be two positive numbers. Analogously to the famous Stieltjes problem from electrostatics the authors study the extremal problem for the function \[ T_n(x_1,\dots,x_n,y_1,\dots,y_{n-1})= \prod_{i=1}^n (1-x_i)^p \prod_{j=1}^n\prod_{k=1}^{n-1}|x_j-y_k| \prod_{l=1}^{n-1}(1+y_l)^q \] over all configurations satisfying \[ -1=y_0<x_1<y_1<\dots<y_{n-1}<x_{n}<y_n=1. \] For the case of an even number of points the expression and the interlacing property are similar. A system \(X=\{x_j\}\), \(Y=\{y_k\}\) for which the maximum is attained is called a \((p,q)\)-Menke system for \([-1,1]\). It turns out that the Menke points are related to the zeros and extrema of Jacobi polynomials. Here is a typical result in that direction. Theorem. The points \(X\) of the Menke system are the zeros and \(Y\) are the extrema of the Jacobi polynomials \(P_n^{(\alpha,\beta)}\) on \([-1,1]\), where \(\alpha=p-1\), \(\beta=q-1\). Similar results are established for the half line and the whole real line.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 5650059
zbMATH Keywords: Fekete points
zbMATH Keywords: logarithmic energy
zbMATH Keywords: Menke points
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: extremal problem
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: j.cam.2009.02.059
OpenAlex ID: W2044271372
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DOI: 10.1016/J.CAM.2009.02.059
description: scientific article; zbMATH DE number 5650059