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Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

Ameur, Yacin LU and Ortega-Cerda, Joaquim (2012) In Journal of Functional Analysis 263(7). p.1825-1861
Abstract
Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i<j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -> infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of... (More)
Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i<j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -> infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials. (C) 2012 Elsevier Inc. All rights reserved. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Weighted Fekete set, Droplet, Equidistribution, Concentration operator, Correlation kernel
in
Journal of Functional Analysis
volume
263
issue
7
pages
1825 - 1861
publisher
Elsevier
external identifiers
  • wos:000307906800003
  • scopus:84864839327
ISSN
0022-1236
DOI
10.1016/j.jfa.2012.06.01
language
English
LU publication?
yes
id
ab4efc7c-b8ba-4bfe-a4f9-973556b5489f (old id 3146789)
date added to LUP
2012-11-26 09:40:24
date last changed
2017-07-02 04:05:32
@article{ab4efc7c-b8ba-4bfe-a4f9-973556b5489f,
  abstract     = {Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i&lt;j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -&gt; infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials. (C) 2012 Elsevier Inc. All rights reserved.},
  author       = {Ameur, Yacin and Ortega-Cerda, Joaquim},
  issn         = {0022-1236},
  keyword      = {Weighted Fekete set,Droplet,Equidistribution,Concentration operator,Correlation kernel},
  language     = {eng},
  number       = {7},
  pages        = {1825--1861},
  publisher    = {Elsevier},
  series       = {Journal of Functional Analysis},
  title        = {Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates},
  url          = {http://dx.doi.org/10.1016/j.jfa.2012.06.01},
  volume       = {263},
  year         = {2012},
}