Sine kernel asymptotics for a class of singular measures

Jonathan Breuer

doi:10.1016/j.jat.2011.05.006

Sine kernel asymptotics for a class of singular measures

Jonathan Breuer^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We construct a family of measures on R that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.

Original language	English
Pages (from-to)	1478-1491
Number of pages	14
Journal	Journal of Approximation Theory
Volume	163
Issue number	10
DOIs	https://doi.org/10.1016/j.jat.2011.05.006
State	Published - Oct 2011

Bibliographical note

Funding Information:
We thank Yoram Last and Barry Simon for useful discussions. We also thank the referee for useful suggestions which improved the presentation of our results. This research was supported by The Israel Science Foundation (Grant no. 1105/10 ).

Keywords

Christoffel-Darboux kernel
Singular continuous measure
Universality

Access to Document

10.1016/j.jat.2011.05.006

Cite this

@article{83a9cecb88b5411e82198215e0ea1cbd,

title = "Sine kernel asymptotics for a class of singular measures",

abstract = "We construct a family of measures on R that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.",

keywords = "Christoffel-Darboux kernel, Singular continuous measure, Universality",

author = "Jonathan Breuer",

note = "Funding Information: We thank Yoram Last and Barry Simon for useful discussions. We also thank the referee for useful suggestions which improved the presentation of our results. This research was supported by The Israel Science Foundation (Grant no. 1105/10 ). ",

year = "2011",

month = oct,

doi = "10.1016/j.jat.2011.05.006",

language = "אנגלית",

volume = "163",

pages = "1478--1491",

journal = "Journal of Approximation Theory",

issn = "0021-9045",

publisher = "Academic Press Inc.",

number = "10",

}

TY - JOUR

T1 - Sine kernel asymptotics for a class of singular measures

AU - Breuer, Jonathan

N1 - Funding Information: We thank Yoram Last and Barry Simon for useful discussions. We also thank the referee for useful suggestions which improved the presentation of our results. This research was supported by The Israel Science Foundation (Grant no. 1105/10 ).

PY - 2011/10

Y1 - 2011/10

N2 - We construct a family of measures on R that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.

AB - We construct a family of measures on R that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.

KW - Christoffel-Darboux kernel

KW - Singular continuous measure

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=80051576393&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2011.05.006

DO - 10.1016/j.jat.2011.05.006

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:80051576393

SN - 0021-9045

VL - 163

SP - 1478

EP - 1491

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 10

ER -

Sine kernel asymptotics for a class of singular measures

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this