Matematicheskie Zametki
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 2021, Volume 110, Issue 4, Pages 592–597 (Mi mz13157)  

Two Examples Related to Properties of Discrete Measures

S. P. Suetin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $\mu$ with $\operatorname{supp}{\mu}=[-1,1]$ whose logarithmic potential is continuous on $[-1,1]$, there exists a (discrete) measure $\sigma=\sigma(\mu)$ with $\operatorname{supp}{\sigma}=[-1,1]$ such that the corresponding orthogonal polynomials $P_n(x;\sigma)=x^n+\dotsb$ satisfy the condition $(1/n)\chi(P_n( \cdot ;\sigma))\xrightarrow{*}\mu$, $n\to\infty$, where $\chi( \cdot )$ is the measure counting the zeros of a polynomial. The proof of the existence of such a measure $\sigma$ is based on properties of weighted Leja points. In the second part, an example of a compact set and a sequence of discrete measures supported on it with a special property is given. Namely, the sequence of measures converges in the $*$-weak topology to the equilibrium measure on the compact set, but the corresponding sequence of logarithmic potentials converges in capacity to the equilibrium potential in no neighborhood of this compact set.

Keywords: orthogonal polynomial, discrete measure, logarithmic potential, convergence in capacity.

Funding Agency Grant Number
Russian Science Foundation 19-11-00316
This work was supported by the Russian Science Foundation under grant 19-11-00316.


DOI: https://doi.org/10.4213/mzm13157

Full text: PDF file (452 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Mathematical Notes, 2021, 110:4, 578–582

Bibliographic databases:

UDC: 517.5
Received: 22.05.2021

Citation: S. P. Suetin, “Two Examples Related to Properties of Discrete Measures”, Mat. Zametki, 110:4 (2021), 592–597; Math. Notes, 110:4 (2021), 578–582

Citation in format AMSBIB
\Bibitem{Sue21}
\by S.~P.~Suetin
\paper Two Examples Related to Properties of Discrete Measures
\jour Mat. Zametki
\yr 2021
\vol 110
\issue 4
\pages 592--597
\mathnet{http://mi.mathnet.ru/mz13157}
\crossref{https://doi.org/10.4213/mzm13157}
\transl
\jour Math. Notes
\yr 2021
\vol 110
\issue 4
\pages 578--582
\crossref{https://doi.org/10.1134/S0001434621090285}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000711049900028}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85118215945}


Linking options:
  • http://mi.mathnet.ru/eng/mz13157
  • https://doi.org/10.4213/mzm13157
  • http://mi.mathnet.ru/eng/mz/v110/i4/p592

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:80
    References:4
    First page:5

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021