Skip to main content
SpringerLink
Log in

Nevanlinna–Pick Problem for Stieltjes Matrix Functions

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We consider the Nevanlinna–Pick interpolation problem for Stieltjes matrix functions. We obtain two criteria for the indeterminacy of the Nevanlinna–Pick problem with infinitely many interpolation nodes. In the indeterminate case, we describe the general solution of the Nevanlinna–Pick problem in terms of fractional-linear transformations.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Yu. M. Dyukarev and V. É. Katsnel'son, “Multiplicative and additive Stieltjes classes of analytic matrix functions and related in-terpolation problems,” Teor. Funkts. Funkts. Anal. Prilozhen.,Issue 36, 13–27 (1981).

    Google Scholar 

  2. D. Aplay, J. Ball, I. Gohberg, and L. Rodman, “Interpolation in the Stieltjes class,” Lin. Algebra Appl.,485–521 (1994).

  3. Yu. M. Dyukarev, “General scheme of the solution of interpolation problems in the Stieltjes class based on consistent integral representations of pairs of nonnegative operators. I,” Mat. Fiz. Anal. Geometr.,6, Nos. 1–2, 30–54 (1999).

    Google Scholar 

  4. I. V. Kovalishina and V. P. Potapov, “Indefinite metric in the Nevanlinna–Pick problem,” Dokl. Akad. Nauk Arm. SSR, 59, Issue 1, 17–22 (1974).

    Google Scholar 

  5. I. V. Kovalishina, “Analytic theory of one class of interpolation problems,” Izv. Akad. Nauk SSSR, Ser. Mat.,47, No. 3, 455–497 (1983)

    Google Scholar 

  6. M. G. Krein and A. A. Nudel'man, Markov Moment Problem and Extremal Problems[in Russian], Nauka, Moscow (1973).

    Google Scholar 

  7. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space[in Russian], Vyshcha Shkola, Kharkov (1977).

    Google Scholar 

  8. Yu. M. Dyukarev, “Integral representations of a pair of nonnegative operators and interpolation problems on the Stieltjes class,” Oper. Theor. Adv. Appl.,95, 165–184 (1997).

    Google Scholar 

  9. Yu. M. Dyukarev and A. E. Choque Rivero, “Power moment problem on a compact interval,” Mat. Zametki, 69, Issue 2, 200–213 (2001).

    Google Scholar 

  10. V. P. Potapov, “Fractional-linear transformations of matrices,” in: V. A. Marchenko (editor), Investigations in the Theory of Op-erators and Their Applications[in Russian], Naukova Dumka, Kiev (1979), pp. 75–97.

  11. W. F. Donoghue, Monotone Matrix Functions and Analytic Continuation,Springer, New York (1974).

    Google Scholar 

  12. S. A. Orlov, “Nested matrix disks analytically dependent on a parameter and theorems on the invariance of ranks of radii of limiting matrix disks,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 3, 593–644 (1976).

    Google Scholar 

  13. V. P. Potapov, “On the theory of Weyl matrix disks,” in: V. A. Marchenko (editor), Functional Analysis and Applied Mathematics[in Russian], Naukova Dumka, Kiev (1982), pp. 113–121.

    Google Scholar 

  14. V. P. Potapov, “Multiplicative structure of J-expanding matrix functions,” Tr. Mosk. Mat. Obshch.,4, 125–236 (1955).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kharkov University, Kharkov

    Yu. M. Dyukarev

Authors
  1. Yu. M. Dyukarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dyukarev, Y.M. Nevanlinna–Pick Problem for Stieltjes Matrix Functions. Ukrainian Mathematical Journal 56, 446–465 (2004). https://doi.org/10.1023/B:UKMA.0000045689.16180.31

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:UKMA.0000045689.16180.31

Keywords

Search

Navigation