A general discrepancy theorem
type: Item
instance of: scholarly article
title: A general discrepancy theorem
zbMATH Open document ID: 0797.30032
DOI: 10.1007/BF02559485
published in: Arkiv för Matematik
publication date: 1994-04-24T00:00:00Z
review text: The discrepancy \(D [\sigma]\) of a signed measure \(\sigma\) on a Jordan arc (curve) \(E\) is defined by \(D [\sigma] : = \sup | \sigma (J) |\), the supremum being taken over all subarcs \(J\) of \(E\). If \(\{\nu_ n\}\) is a sequence of Borel measures on \(E\) the discrepancy \(D [\nu_ n - \mu]\) serves as a measurement on the weak-star convergence of \(\{\nu_ n\}\) to a Borel measure \(\mu\). Main result: Let \(E\) be a Jordan arc (curve) of class \(C^{1+}\). Let \(p_ n\) be a monic polynomial of degree \(n\) with zeros \(z_ i\) in \(E\), \(1 \leq i \leq n\), such that (a) \(\max_{z \in E} | p_ n (z) | \leq A_ n (\text{cap} E)^ n\), (b) \(| p_ n' (z_ i) | \geq B_ n^{-1} (\text{cap} E)^ n\), (c) \(C_ n : = \max (A_ n, B_ n,n) \leq e^{n/e}\). Finally let \(\nu_ n\) denote the measure which associates the mass \(1/n\) with each of the zeros \(z_ i\). Then \[ D [\nu_ n - \mu_ E] \leq c {\log C_ n \over n} \log {n \over \log C_ n}, \] where \(c\) is a positive constant depending only on \(E\) and \(\mu_ E\) is the equilibrium measure of \(E\). The main result is applied to get estimates of the distribution of Fekete points, extreme points of polynomials of best approximation and zeros of orthogonal polynomials on the unit circle and on compact intervals.
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 556411
zbMATH Keywords: discrepancy of a signed measure
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: distribution of Fekete points
author: H.-P. Blatt
author: H. N. Mhaskar
reviewed by: Józef Siciak
MaRDI profile type: Publication
cites work: On the distribution of simple zeros of polynomials
cites work: Erdős-Turán theorems on a system of Jordan curves and arcs
cites work: The distribution of extreme points in best complex polynomial approximation
cites work: Q5618011
cites work: On Orthogonal Polynomials with Regularly Distributed Zeros
cites work: On the uniformly-dense distribution of certain sequences of points
cites work: On interpolation. III: Interpolatory theory of polynomials
cites work: On the distribution of roots of polynomials
cites work: A class of orthogonal polynomials
cites work: Q3251576
cites work: Q5622304
cites work: Sur l'approximation de la représentation conforme par la méthode des points extrémaux de M. Leja
cites work: Q5666837
cites work: Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle
cites work: Q5531436
cites work: On the distribution of zeros of polynomials orthogonal on the unit circle
cites work: Q3032769
cites work: Polynome und konforme Abbildung
cites work: Über die Verteilung der Fekete-Punkte
cites work: ON THE ASYMPTOTICS OF THE RATIO OF ORTHOGONAL POLYNOMIALS. II
cites work: Q3473617
cites work: Q5587306
cites work: Estimates of mass distributions from their potentials and energies
cites work: Distribution of simple zeros of polynomials
cites work: On removable singularities of an analytic function of several complex variables
cites work: On the Distortion in Conformal Mapping of Variable Domains
cites work: Extremal polynomials associated with a system of curves in the complex plane
cites work: Q4143689
full work available at URL: https://doi.org/10.1007/bf02559485
OpenAlex ID: W2009204583
description: scientific article; zbMATH DE number 556411