Skip to main content
Log in

Co-Recursivity and karlin-McGregor duality for indeterminate moment problems

  • Published:
Constructive Approximation Aims and scope

Abstract

We analyze co-recursivity for indeterminate Hamburger moment problems and the duality transformation of Karlin and McGregor for indeterminate Stieltjes moment problems. In both cases the transformed Nevanlinna matrix is given and the Nevanlinna extremal measures are discussed. An example involving associated polynomials, relevant for a quartic birth and death process, is worked out.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. N. I. Akhiezer (1965):The Classical Moment Problem. Edinburgh: Oliver and Boyd.

    Google Scholar 

  2. C. Berg (1994):Markov’s theorem revisited. J. Approx. Theory,78:260–275.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Berg, H. L. Pedersen (1994):On the order and type of the entire functions associated with an indeterminate Hamburger problem. Arkiv. Math.,32:1–11.

    MATH  MathSciNet  Google Scholar 

  4. C. Berg, G. Valent (1994):The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes. Methods Appl. Anal.,1:169–209.

    MATH  MathSciNet  Google Scholar 

  5. T. S. Chihara (1957):On co-recursive orthogonal polynomials. Proc. Amer. Math. Soc.,8:899–905.

    Article  MATH  MathSciNet  Google Scholar 

  6. T. S. Chihara (1982):Indeterminate symmetric moment problems. J. Math. Anal. Appl.,85:331–346.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. S. Chihara, M. E. H. Ismail (1993):Extremal measures for a system of orthogonal polynomials. Constr. Approx.9:111–119.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. E. H. Ismail, D. Masson (1994):q-Hermite polynomials, biorthogonal functions and q-beta integrals. Trans. Amer. Math. Soc.,346:63–116.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Karlin, J. McGregor (1957):The classification of birth and death processes. Trans. Amer. Math. Soc.86:366–401.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Karlin, J. McGregor (1958):The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc.,85:489–546.

    Article  MathSciNet  Google Scholar 

  11. W. Magnus, F. Oberhettinger, R. P. Soni (1966): Formulas and Theorems of the Special Functions of Mathematical Physics, New York, Springer-Verlag.

    Google Scholar 

  12. A. I. Markushevich (1965): Theory of Functions of a Complex Variable, volume 2. New York: Chelsea.

    Google Scholar 

  13. H. L. Pedersen (1995):Stieltjes moment problems and the Friedrichs extension of a positive definite operator. J. Approx. Theory83:289–307.

    Article  MATH  MathSciNet  Google Scholar 

  14. L.J. Slater (1966): Generalized Hypergeometric Series. Cambridge: Cambridge University Press.

    Google Scholar 

  15. J. A. Shohat, J. D. Tamarkin (1950): The Problem of Moments. Mathematical Surveys, Volume 1, Amer. Math. Soc., Providence, RI.

    Google Scholar 

  16. W. Van Assche (1991):Orthogonal polynomials, associated polynomials and functions of the second kind. J. Comput. Appl. Math.37:237–249.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Valent (1993):Orthogonal polynomials for a quartic birth and death process. J. Comp. Appl. Math.49:281–288.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Valent (1994):Asymptotic analysis of some associated orthogonal polynomials connected with elliptic functions. SIAM J. Math. Anal.,25:749–775.

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Valent (1996):Exact solutions of a quartic birth and death process and related orthogonal polynomials. J. Comput. Appl. Math.67:103–127.

    Article  MATH  MathSciNet  Google Scholar 

  20. E. T. Whittaker, G. N. Watson (1965): A Course of Modern Analysis. Cambridge: Cambridge University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by Mourad Ismail.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Valent, G. Co-Recursivity and karlin-McGregor duality for indeterminate moment problems. Constr. Approx 12, 531–553 (1996). https://doi.org/10.1007/BF02437507

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02437507

AMS classification

Key words and phrases

Navigation