### Abstract

**R**, which possesses all power moments, is N-extremal if the space of all polynomials is dense in L

^{2}(μ). If, in addition, μ 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 supp μ; for example, if supp μ={n

^{σ}log n: n∈N} with some σ>2, then shifts of mass points behaving asymptotically like, e.g. n

^{σ-2}[log log n]

^{-2}are permitted.

Language | English |
---|---|

Pages | 69-75 |

Number of pages | 7 |

Journal | Methods of Functional Analysis and Topology |

Volume | 21 |

Issue number | 1 |

Publication status | Published - Mar 2015 |

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### Keywords

- Hamburger moment problem
- N-extremal measure
- perturbation of support

### Cite this

*Methods of Functional Analysis and Topology*,

*21*(1), 69-75.

}

*Methods of Functional Analysis and Topology*, vol. 21, no. 1, pp. 69-75.

**Stability of N-extremal measures.** / Langer, Matthias; Woracek, Harald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stability of N-extremal measures

AU - Langer, Matthias

AU - Woracek, Harald

PY - 2015/3

Y1 - 2015/3

N2 - A positive Borel measure μ on R, which possesses all power moments, is N-extremal if the space of all polynomials is dense in L2(μ). If, in addition, μ 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 supp μ; for example, if supp μ={nσ log n: n∈N} with some σ>2, then shifts of mass points behaving asymptotically like, e.g. nσ-2[log log n]-2 are permitted.

AB - A positive Borel measure μ on R, which possesses all power moments, is N-extremal if the space of all polynomials is dense in L2(μ). If, in addition, μ 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 supp μ; for example, if supp μ={nσ log n: n∈N} with some σ>2, then shifts of mass points behaving asymptotically like, e.g. nσ-2[log log n]-2 are permitted.

KW - Hamburger moment problem

KW - N-extremal measure

KW - perturbation of support

UR - http://www.imath.kiev.ua/~mfat/

UR - http://www.imath.kiev.ua/~mfat/html/papers/2015/1/lan-wo/art.pdf

M3 - Article

VL - 21

SP - 69

EP - 75

JO - Methods of Functional Analysis and Topology

T2 - Methods of Functional Analysis and Topology

JF - Methods of Functional Analysis and Topology

SN - 1029-3531

IS - 1

ER -