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.


Full Text





Article Information

TitleStability of N-extremal measures
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 69-75
MathSciNet MR3407921
zbMATH 06533468
MilestonesReceived 21/03/2014; Revised 15/06/2014
CopyrightThe 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


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley



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},
}


References

  1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.  MathSciNet
  2. Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954.  MathSciNet
  3. Ch. Berg, J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, vi, 99-114.  MathSciNet
  4. C. Berg and H. L. Pedersen, Nevanlinna extremal measures and zeros of entire functions. Problem 12.5, Linear and Complex Analysis Problem Book II, Lecture Notes in Mathematics, vol. 1574, 1994, pp. 89-91.
  5. Alexander Borichev, Mikhail Sodin, The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line, J. Anal. Math. 76 (1998), 219-264.  MathSciNet CrossRef
  6. Hans Ludwig Hamburger, Hermitian transformations of deficiency-index $(1,1)$, Jacobi matrices and undetermined moment problems, Amer. J. Math. 66 (1944), 489-522.  MathSciNet
  7. Paul Koosis, Mesures orthogonales extremales pour lapproximation ponderee par des polynomes, C. R. Acad. Sci. Paris S\er. I Math. 311 (1990), no. 9, 503-506.  MathSciNet
  8. Ja. B. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1980.  MathSciNet
  9. Pierre Lelong, Lawrence Gruman, Entire functions of several complex variables, Springer-Verlag, Berlin, 1986.  MathSciNet CrossRef
  10. Matthias Langer, Harald Woracek, Stability of the derivative of a canonical product, Complex Anal. Oper. Theory 8 (2014), no. 6, 1183-1224.  MathSciNet CrossRef
  11. Henrik L. Pedersen, Logarithmic order and type of indeterminate moment problems. II, J. Comput. Appl. Math. 233 (2009), no. 3, 808-814.  MathSciNet CrossRef
  12. J. A. Shohat, J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York, 1943.  MathSciNet


All Issues