Skip to main content

Jacobi Matrices and de Branges Spaces

  • Reference work entry
  • First Online:
Operator Theory
  • 4416 Accesses

Abstract

This is a short survey on relationships between Jacobi matrices, de Branges spaces, and canonical systems.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, Edinburgh (1965)

    MATH  Google Scholar 

  2. Baranov, A.: Polynomials in the de Branges spaces of entire functions. Ark. Mat. 44, 16–38 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baranov, A., Belov, Yu., Borichev, A.: Strong M-basis property for systems of reproducing kernels in de Branges spaces. arXiv:1309.6915v1 [math.CV] (2013)

    Google Scholar 

  4. Berg, C., Valent, G.: The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes. Methods Appl. Anal. 1(2), 169–209 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Borichev, A., Sodin, M.: The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line. J. Anal. Math. 76, 219–264 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borichev, A., Sodin, M.: Weighted polynomial approximation and the Hamburger moment problem. In: Complex Analysis and Differential Equations. Proceedings of the Marcus Wallenberg Symposium in Honor of Matts Essén, Uppsala University (1998)

    Google Scholar 

  7. Christiansen, J.: Indeterminate moment problems within the Askey-scheme. Ph.D. thesis, University of Copenhagen (2004)

    Google Scholar 

  8. de Branges, L.: Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs (1968)

    MATH  Google Scholar 

  9. Gilewicz, J., Leopold, E., Valent, G.: New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes. J. Comput. Appl. Math. 178, 235–245 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hassi, S., de Snoo, H., Winkler, H.: Boundary-value problems for two-dimensional canonical systems. Integr. Equ. Oper. Theory 36(4), 445–479 (2000)

    Article  MATH  Google Scholar 

  11. Hayman, W. K.: On the zeros of a q-Bessel function. Contemp. Math. 382, 205–216 (2005)

    Article  MathSciNet  Google Scholar 

  12. Ismail, M.E.H., Masson, D.R.: q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals. Trans. AMS 346, 63–116 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Ismail, M., Valent, G., Yoon, G.J.: Some orthogonal polynomials related to elliptic functions. J. Approx. Theory 112(2), 251–278 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kac, I.S.: Inclusion of Hamburger’s power moment problem in the spectral theory of canonical systems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999, in Russian). Issled. po Linein. Oper. i Teor. Funkts. 27, 147–171. Translation in J. Math. Sci. (New York) 110(5), 2991–3004 (2002)

    Google Scholar 

  15. Karlin, S., McGregor, J.L.: The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. AMS 85, 489–546 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  16. Moak, D.S.: The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81, 20–47 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reming, C.: Schrödinger operators and de Branges spaces. J. Funct. Anal. 196(2), 323–394 (2002)

    Article  MathSciNet  Google Scholar 

  18. Woracek, H.: de Branges spaces and growth aspects. In: Alpay, D. (ed.) Operator Theory, chapter 21, pp. 489–524, Springer, Basel (2015). doi: 10.1007/978-3-0348-0692-3_7

    Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Austrian Science Fund (FWF) project I 1536–N25, and the Russian Foundation for Basic Research, Grants 13-01-91002-ANF and 12-01-00215. The author is indebted to Yu. Belov for explanations concerning [11].

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roman Romanov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Romanov, R. (2015). Jacobi Matrices and de Branges Spaces. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_9

Download citation

Publish with us

Policies and ethics