A local discrepancy theorem
type: Item
instance of: scholarly article
title: A local discrepancy theorem
zbMATH Open document ID: 1013.42017
DOI: 10.1016/S0019-3577(01)80003-3
author: V. V. Andrievskii
author: H.-P. Blatt
author: H. N. Mhaskar
published in: Indagationes Mathematicae. New Series
publication date: 2003-06-22T00:00:00Z
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 33C47
zbMATH DE Number: 1663806
zbMATH Keywords: logarithmic potential
zbMATH Keywords: counting measure
zbMATH Keywords: discrepancy
zbMATH Keywords: weak-star convergence
zbMATH Keywords: signed measures
zbMATH Keywords: Pollaczek polynomials
MaRDI profile type: Publication
cites work: A discrepancy theorem on quasiconformal curves
cites work: Erdős-Turán type theorems on quasiconformal curves and arcs
cites work: Local discrepancy theorems for the distributions of zeros of polynomials
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: A general discrepancy theorem
cites work: A discrepancy theorem concerning polynomials of best approximation in \(L^ p_ w [-1, 1]\)
cites work: The distribution of extreme points in best complex polynomial approximation
cites work: On the distribution of roots of polynomials
cites work: Q5555819
cites work: Sequences of analytic functions and their zeros
cites work: Q3803598
cites work: Q4857769
cites work: Orthogonal polynomials
cites work: Q4360405
cites work: Q4195371
cites work: Estimates of mass distributions from their potentials and energies
cites work: Distribution of simple zeros of polynomials
cites work: Q3259806
review text: This excellent paper continues important work on discrepancy theory with applications on Pollaczek polynomials. The main result gives an estimate using local discrepancy (for a comparable result using global dicrepancy cf. \textit{H.-P. Blatt} and \textit{H. N. Mhaskar}, Ark. Mat. 31, No. 2, 219-246 (1993; Zbl 0797.30032)]. NEWLINENEWLINENEWLINELet \(\sigma\) be a signed Borel measure on \(I=[-1,1]\) and let \(J=[a,b]\subset I\) be a subinterval. The local discrepancy of \(\sigma\) on \(J\) is defined by NEWLINE\[NEWLINE D_J[\sigma]=\sup_{[c,d]\subset J} |\sigma([c,d])|.NEWLINE\]NEWLINE The main result concerns the signed measure \(\sigma=\sigma^{+}-\sigma^{-}\) with \(\sigma^{\pm}\) positive unit Borel measures on \(I\), \(\sigma^{+}(J)\leq M(b-a)^{\gamma}\) for certain \(M>0, 0<\gamma\leq 1\) which are independent of \(a, b\): NEWLINE\[NEWLINE D_J[\sigma]\leq 4\varepsilon\left\{2+\log{\left(1+{b-a\over \delta}\right)}\right\} + 16\delta\log{{1\over \delta}}+2M\delta^{\gamma} NEWLINE\]NEWLINE for all \(\delta\) with \(0<\delta<{1\over 2}\) and \(\varepsilon\) defined by NEWLINE\[NEWLINE \varepsilon=\max_{z\in B(\delta,J)} |U^{\sigma}(z)|. NEWLINE\]NEWLINE Here \(B(\delta,J)=\{z=x+iy\mid a-\delta\leq x\leq b+\delta\), \(|y|\geq \delta^2\}\) and NEWLINE\[NEWLINE U^{\sigma}(z)=\int\log{{1\over |z-t|}}d\sigma(t) NEWLINE\]NEWLINE is the logarithmic potential induced by \(\sigma\). NEWLINENEWLINENEWLINEIntroducing the generalized Pollaczek weight (GP) \(W\) by supp\((W)=I\) and NEWLINE\[NEWLINE W(x)=\exp{\left(-{\sqrt{1-x^2}\over |P(x)|}\right)},\quad -1\leq x\leq 1, NEWLINE\]NEWLINE where \(P\) is a polynomial having all its zeros simple and on \(I\), application of the local discrepancy bound leads to: NEWLINENEWLINENEWLINELet \(\tau\) be a positive Borel measure on \(I\), \(W\) a GP-weight and let the Radon-Nikodým derivative \(\tau'\) satisfy \(\tau'(x)\geq W(x)\) a.e. in \(I\). Then: NEWLINE\[NEWLINE D_{[a,b]}[\nu_n(\tau)-\mu]\leq c{(\log{n})^2\over n} (n\geq 2) NEW…
description: scientific article; zbMATH DE number 1663806