Open Access

# 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.

### 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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