Stability of N-extremal measures

Abstract

A positive Borel measure $\mu$ on $\mathbb R$, which possesses all power moments, is N-extremal if the space of all polynomials is dense in $L^2(\mu)$. If, in addition, $\mu$ generates an indeterminate Hamburger moment problem, then it is discrete. It is known that the class of N-extremal measures that generate an indeterminate moment problem is preserved when a finite number of mass points are moved (not ``removed''!). We show that this class is preserved even under change of infinitely many mass points if the perturbations are asymptotically small. Thereby ``asymptotically small'' is understood relative to the distribution of ${\rm supp}\mu$; for example, if ${\rm supp}\mu=\{n^\sigma\log n:\,n\in\mathbb N\}$ with some $\sigma>2$, then shifts of mass points behaving asymptotically like, e.g. $n^{\sigma-2}[\log\log n]^{-2}$ are permitted.

Key words: Hamburger moment problem, N-extremal measure, perturbation of support.

Full Text

Article Information

Title	Stability of N-extremal measures
Source	Methods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 69-75
MathSciNet	MR3407921
zbMATH	06533468
Milestones	Received 21/03/2014; Revised 15/06/2014
Copyright	The Author(s) 2015 (CC BY-SA)

Authors Information

Matthias Langer
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

Harald Woracek
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrase 8--10/101, 1040 Wien, Austria

Citation Example

Matthias Langer and Harald Woracek, Stability of N-extremal measures, Methods Funct. Anal. Topology 21 (2015), no. 1, 69-75.

BibTex

@article {MFAT743,
    AUTHOR = {Langer, Matthias and Woracek, Harald},
     TITLE = {Stability of N-extremal measures},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {69-75},
      ISSN = {1029-3531},
  MRNUMBER = {MR3407921},
 ZBLNUMBER = {06533468},
       URL = {http://mfat.imath.kiev.ua/article/?id=743},
}

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