Open Access

On a class of generalized Stieltjes continued fractions

     Article (.pdf)

Abstract

With each sequence of real numbers ${\mathbf s}=\{s_j\}_{j=0}^\infty$ two kinds of continued fractions are associated, - the so-called $P-$fraction and a generalized Stieltjes fraction that, in the case when ${\mathbf s}=\{s_j\}_{j=0}^\infty$ is a sequence of moments of a probability measure on $\mathbb R_+$, coincide with the $J-$fraction and the Stieltjes fraction, respectively. A subclass $\mathcal H^{reg}$ of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For ${\mathbf s}\in\mathcal H^{reg}$ the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.

Key words: Moment problem, continued fraction, generalized Stieltjes fraction, Darboux transformation, generalized Jacobi matrix, triangular factorization, unwrapping transformation.


Full Text





Article Information

TitleOn a class of generalized Stieltjes continued fractions
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 315-335
MathSciNet MR3469531
zbMATH 06630277
MilestonesReceived 02/02/2015; Revised 05/03/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

Vladimir Derkach
Department of Mathematics, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka,Ivano-Frankivsk, 76018, Ukraine

Ivan Kovalyov
Department of Mathematics, Dragomanov National Pedagogical University, 9 Pirogova, Kyiv, 01601, Ukraine


Citation Example

Vladimir Derkach and Ivan Kovalyov, On a class of generalized Stieltjes continued fractions, Methods Funct. Anal. Topology 21 (2015), no. 4, 315-335.


BibTex

@article {MFAT781,
    AUTHOR = {Derkach, Vladimir and Kovalyov, Ivan},
     TITLE = {On a class of generalized Stieltjes continued fractions},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {4},
     PAGES = {315-335},
      ISSN = {1029-3531},
  MRNUMBER = {MR3469531},
 ZBLNUMBER = {06630277},
       URL = {http://mfat.imath.kiev.ua/article/?id=781},
}


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar


Export article

Save to Mendeley


References

  1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.  MathSciNet
  2. George A. Baker Jr., Peter Graves-Morris, Pade approximants. Part I, Addison-Wesley Publishing Co., Reading, Mass., 1981.  MathSciNet
  3. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
  4. M. I. Bueno, F. Marcellan, Darboux transformation and perturbation of linear functionals, Linear Algebra Appl. 384 (2004), 215-242.  MathSciNet CrossRef
  5. M. Derevyagin, On the Schur algorithm for indefinite moment problem, Methods Funct. Anal. Topology 9 (2003), no. 2, 133-145.  MathSciNet
  6. Maxim Derevyagin, On the relation between Darboux transformations and polynomial mappings, J. Approx. Theory 172 (2013), 4-22.  MathSciNet CrossRef
  7. Maxim Derevyagin, Spectral theory of the $G$-symmetric tridiagonal matrices related to Stahls counterexample, J. Approx. Theory 191 (2015), 58-70.  MathSciNet CrossRef
  8. Maxim Derevyagin, Vladimir Derkach, Spectral problems for generalized Jacobi matrices, Linear Algebra Appl. 382 (2004), 1-24.  MathSciNet CrossRef
  9. Maxim Derevyagin, Vladimir Derkach, Darboux transformations of Jacobi matrices and Pade approximation, Linear Algebra Appl. 435 (2011), no. 12, 3056-3084.  MathSciNet CrossRef
  10. Vladimir Derkach, On indefinite moment problems and resolvent matrices of Hermitian operators in Krei n spaces, Math. Nachr. 184 (1997), 135-166.  MathSciNet CrossRef
  11. V. A. Derkach, M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1-95.  MathSciNet CrossRef
  12. I. Gohberg, P. Lancaster, L. Rodman, Matrix polynomials, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.  MathSciNet
  13. Olga Holtz, Mikhail Tyaglov, Structured matrices, continued fractions, and root localization of polynomials, SIAM Rev. 54 (2012), no. 3, 421-509.  MathSciNet CrossRef
  14. I. S. Kac and M. G. Krein, $R$-functions - analytic functions mapping the upper halfplane into itself, Supplement to the Russian edition of F. V. Atkinson, Discrete and Continuous Boundary Problems, Mir, Moscow, 1968. (Russian); English transl. Amer. Math. Soc. Transl. Ser. 2 103 (1974), 1-18.
  15. M. Kaltenback, H. Winkler, H. Woracek, Symmetric relations of finite negativity, in: Operator theory in Krein spaces and nonlinear eigenvalue problems, Birkhauser, Basel, 2006.  MathSciNet CrossRef
  16. Ivan Kovalyov, Darboux transformation of generalized Jacobi matrices, Methods Funct. Anal. Topology 20 (2014), no. 4, 301-320.  MathSciNet
  17. M. G. Krein, H. Langer, Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _\kappa $ zusammenhangen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236.  MathSciNet
  18. M. G. Krein, Heinz Langer, On some extension problems which are closely connected with the theory of Hermitian operators in a space $\Pi _\kappa $. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I, Beitrage Anal. (1979), no. 14, 25-40 (loose errata).  MathSciNet
  19. M. G. Krein, H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space $\Pi _\kappa $, Acta Sci. Math. (Szeged) 43 (1981), no. 1-2, 181-205.  MathSciNet
  20. M. G. Krein, A. A. Nudel′man, The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977.  MathSciNet
  21. Arne Magnus, Expansion of power series into $P$-fractions, Math. Z. 80 (1962), 209-216.  MathSciNet
  22. Franz Peherstorfer, Finite perturbations of orthogonal polynomials, J. Comput. Appl. Math. 44 (1992), no. 3, 275-302.  MathSciNet CrossRef
  23. Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82-203.  MathSciNet CrossRef
  24. T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8 (1894), no. 4, J1-J122.  MathSciNet
  25. H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, New-York, 1967.


All Issues