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Sampling theory for Sturm–Liouville problem with boundary and transmission conditions containing an eigenparameter

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Abstract

In this paper, we derive the sampling theorem associated with a Sturm–Liouville problem which has two points of discontinuity and contains an eigenparameter in a boundary condition and also two transmission conditions. We establish briefly spectral properties of the problem, and then, we prove the sampling theorem associated with the problem.

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References

  1. Paley, R., Wiener, N.: Fourier transforms in the complex domain. Am. Math. Soc. Colloq. Publ. 19 (1934)

  2. Zayed A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  3. Butzer P.L., Schmeisser G.R., Stens L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Nonuniform Sampling, Theory and Practice, pp. 17–21. Kluwer Academic, New York (2001)

    Chapter  Google Scholar 

  4. Levinson, N.: Gap and density theorems. Am. Math. Soc. Colloq. Publ. 26 (1940)

  5. Kramer H.P.: A generalized sampling theorem. J. Math. Phys. 38, 68–72 (1959)

    MATH  Google Scholar 

  6. Everitt W.N., Nasri-Roudsari G., Rehberg J.: A note on the analytic form of the Kramer sampling theorem. Results Math. 34(3–4), 310–319 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Everitt W.N., García A.G., Hernández-Medina M.A.: On Lagrange-type interpolation series and analytic Kramer kernels. Results Math. 51, 215–228 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. García A.G., Littlejohn L.L.: On analytic sampling theory. J. Comput. Appl. Math. 171, 235–246 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Everitt W.N., Nasri-Roudsari G.: Interpolation and sampling theories and linear ordinary boundary value problems. In: Higgins, J.R., Stens, R.L. (eds.) Sampling Theory in Fourier and Signal Analysis: Advanced Topics, Ch. 5, Oxford University Press, Oxford (1999)

    Google Scholar 

  10. Everitt W.N., Schöttler G., Butzer P.L.: Sturm–Liouville boundary value problems and Lagrange interpolation series. J. Rend. Math. Appl. 14, 87–126 (1994)

    MATH  Google Scholar 

  11. Zayed A.I.: On Kramer’s sampling theorem associated with general Sturm–Liouville boundary value problems and Lagrange interpolation. SIAM J. Appl. Math. 51, 575–604 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zayed A.I., Hinsen G., Butzer P.L.: On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm–Liouville problems. SIAM J. Appl. Math. 50, 893–909 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boumenir A., Chanane B.: Eigenvalues of S–L systems using sampling theory. Appl. Anal. 62, 323–334 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Annaby M.H., Bustoz J., Ismail M.E.H.: On sampling theory and basic Sturm–Liouville systems. J. Comput. Appl. Math. 206, 73–85 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boumenir A., Zayed A.I.: Sampling with a string. J. Fourier Anal. Appl. 8, 211–231 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Annaby M.H., Tharwat M.M.: On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. SUT J. Math. 42, 157–176 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Boumenir A.: The sampling method for SL problems with the eigenvalue in the boundary conditions. J. Numer. Func. Anal. Optim. 21, 67–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Annaby M., Freiling G.: A sampling theorem for transforms with discontinuous kernels. Appl. Anal. 83, 1053–1075 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Annaby M.H., Freiling G., Zayed A.I.: Discontinuous boundary-value problems: expansion and sampling theorems. J. Integr. Equ. Appl. 16, 1–23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zayed A.I., García A.G.: Kramer’s sampling theorem with discontinuous kernels. Results Math. 34, 197–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kobayashi M.: Eigenfunction expansions: a discontinuous version. SIAM J. Appl. Math. 50, 910–917 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tharwat, M.M.: Discontinuous Sturm–Liouville problems and associated sampling theories. Abstr. Appl. Anal. doi:10.1155/2011/610232 (2011)

  23. Altınışık N., Kadakal M., Mukhtarov O.Sh.: Eigenvalues and eigenfunctions of discontinuous Sturm Liouville problems with eigenparameter dependent boundary conditions. Acta Math. Hungar. 102, 159–175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kadakal M., Mukhtarov O.Sh.: Sturm Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367–1379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mukhtarov O.Sh., Kadakal M., Altınışık N.: Eigenvalues and eigenfunctions of discontinuous Sturm Liouville problems with eigenparameter in the boundary conditions. Indian J. Pure Appl. Math. 34, 501–516 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Titchmarsh E.C.: Eigenfunctions Expansion Associated with Second Order Differential Equations I. Oxford University Press, London (1962)

    Google Scholar 

  27. Levitan, B.M., Sargjan, I.S.: Introduction to Spectral Theory Self-Adjoint Ordinary Differential Operators. American Mathematical Society, Providence, RI. Translation of Mth., Monographs 39 (1975)

  28. Boas R.P.: Entire Functions. Academic Press, New York (1954)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Arts and Science Faculty, Ondokuz Mayıs University, 55139, Samsun, Turkey

    Fatma Hıra & Nihat Altınışık

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  1. Fatma Hıra
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  2. Nihat Altınışık
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Correspondence to Fatma Hıra.

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Hıra, F., Altınışık, N. Sampling theory for Sturm–Liouville problem with boundary and transmission conditions containing an eigenparameter. Z. Angew. Math. Phys. 66, 1737–1749 (2015). https://doi.org/10.1007/s00033-015-0505-2

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  • DOI: https://doi.org/10.1007/s00033-015-0505-2

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