Log in

Sampling and the Eigenvalues of a Quadratic Pencil

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

We use the sampling method to compute the eigenvalues of a quadratic pencil. To do so we first prove a non standard representation of the eigensolution that allows us to decompose the characteristic function into Paley-Wiener functions. Numerical examples illustrating the computation by the sampling method of real as well as complex eigenvalues are provided at the end.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. M. H. Annaby, and R. M. Asharabi, Bounds for truncation and perturbation errors of nonuniform sampling series, BIT, 56, no. 3, 807–832, 2016.

    Article  MathSciNet  Google Scholar 

  2. M. H. Annaby, and R. M. Asharabi, On sinc-based methods in computing eigenvalues of boundary-value problems, SIAM J. Numer. Anal., 46, no. 2, 671–690, 2008.

    Article  MathSciNet  Google Scholar 

  3. M. H. Annaby and M. M. Tharwat, On computing eigenvalues of second-order linear pencils, IMA Journal of Numerical Analysis, 27, 366–380, 2007.

    Article  MathSciNet  Google Scholar 

  4. N. Bondarenko and G. Freiling, An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line, J. Inverse Ill-Posed Probl., 44, no. 4, 467–495, 2013.

    MathSciNet  MATH  Google Scholar 

  5. A. Boumenir, Eigenvalues of periodic Sturm-Liouville problems by the Shannon-Whittaker sampling, Math. Comp., 68, no. 227, 1057–1066, 1999.

    Article  MathSciNet  Google Scholar 

  6. A. Boumenir, Sampling and eigenvalues of non-self-adjoint Sturm-Liouville problems, SIAM J. Sci. Comput., 23, no. 1, 219–229, 2001.

    Article  MathSciNet  Google Scholar 

  7. J. B. Conway, Functions of One Complex Variable,(v.1), 2nd ed, Graduate Texts in Mathematics - Vol. 11, Springer, 1978.

  8. S. Cox snd E. Zuazua, The rate at which energy decays In a string damped at one end, Communications in Partial Differential Equations, 19, no. 1–2, 213–243, 1994.

    MathSciNet  MATH  Google Scholar 

  9. M. G. Gasymov and G. Guseinov, Determination of a diffusion operator from spectral data, (Russian) Akad. Nauk Azerbaidzhan. SSR Dokl., 37, no. 2, 19–23, 1981.

    MathSciNet  Google Scholar 

  10. M. Hasanov, The spectra of two-parameter quadratic operator pencils, Math. Comput. Modelling, 54, no. 1–2, 742–755, 2011.

    Article  MathSciNet  Google Scholar 

  11. D. Jagerman, Bounds for truncation error of the sampling expansion. SIAM J. Appl. Math., 14, 714–723, 1966.

    Article  MathSciNet  Google Scholar 

  12. H. Koyunbakan, Inverse problem for a quadratic pencil of Sturm-Liouville operator. J. Math. Anal. Appl., 378, no. 2, 549–554, 2011.

    Article  MathSciNet  Google Scholar 

  13. A. G. Kostyuchenko and A. A. Shkalikov, Self-adjoint quadratic operator pencils and elliptic problems, Funct. Anal. Appl., 17, no. 2, 109–128, 1983.

    Article  Google Scholar 

  14. H. Langer, R. Mennicken and C. Tretter, A self-adjoint linear pencil Q-λP of ordinary differential operators, Methods Funct. Anal. Topology, 2, 38–54, 1996.

    MathSciNet  MATH  Google Scholar 

  15. B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surveys, 19, 1–62, 1964.

    Article  Google Scholar 

  16. W. W. Lin and J. N. Wang, Partial pole assignment for the quadratic pencil by output feedback control with feedback designs, Numer. Linear Algebra Appl., 12, no. 10, 967–979, 2005.

    Article  MathSciNet  Google Scholar 

  17. M. Möller, Self-adjoint quadratic operator pencils and applications, Not. S. Afr. Math. Soc., 45, no. 1, 2–13, 2014.

    MathSciNet  Google Scholar 

  18. M. Möller, and V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications, Operator Theory: Advances and Applications, 246, Birkhauser/Springer, Cham, 2015.

    Book  Google Scholar 

  19. I. M. Nabiev, Multiplicity and relative position of the eigenvalues of a quadratic pencil of Sturm-Liouville operators, translation in Math. Notes, 67, no. 3–4, 309–319, 2000.

    Article  MathSciNet  Google Scholar 

  20. V. N. Pivovarchik, Eigenvalues of a certain quadratic pencil of operators, Funct. Anal. Appl., 23, no. 1, 80–81, 1989.

    Article  MathSciNet  Google Scholar 

  21. N. I. Pronska, Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm-Liouville equations, Mat. Stud., 40, no. 1, 38–52, 2013.

    MathSciNet  MATH  Google Scholar 

  22. A. Zayed, Advances in Shannon’s Sampling Theory, CRC Press, 1993.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. Baskaya.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baskaya, E., Boumenir, A. Sampling and the Eigenvalues of a Quadratic Pencil. STSIP 18, 9–22 (2019). https://doi.org/10.1007/BF03549618

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549618

Key words and phrases

2010 AMS Mathematics Subject Classfication

Navigation