Sampling theorems for Sturm-Liouville problem with moving discontinuity points
© Hıra and Altınışık; licensee Springer. 2014
Received: 17 June 2014
Accepted: 30 October 2014
Published: 8 November 2014
In this paper, we investigate the sampling analysis for a new Sturm-Liouville problem with symmetrically located discontinuities which are defined depending on a parameter in a neighborhood of a midpoint of the interval. Also the problem has transmission conditions at these points of discontinuity and includes an eigenparameter in a boundary condition. We establish briefly the relations needed for the derivations of the sampling theorems and construct the Green’s function for the problem. Then we derive sampling representations for the solutions and Green’s functions.
MSC: 34B24, 34B27, 94A20.
In the literature, the Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem and generalization of the WKS sampling theorem (see –) has been investigated extensively (see also –). Sampling theorems associated with Sturm-Liouville problems were investigated in –. Also, – and – are for example works in the direction of sampling analysis for continuous and discontinuous eigenproblems, respectively. The sampling series associated with strings were investigated and one compared them with those associated with Sturm-Liouville problems in . In  the author studied the sampling analysis for the discontinuous Sturm-Liouville problem which had transmission conditions at one point of discontinuity and contained an eigenparameter in two boundary conditions. In the present paper, we derive sampling theorems associated with a new Sturm-Liouville problem with moving discontinuity points. The problem studied in this paper was presented in more detail for the first time in . The problem has symmetrically located discontinuities which are defined depending on a parameter in a neighborhood of the midpoint of the interval and with an eigenparameter appearing in a boundary condition. There are many published works on sampling theorems associated with different types of generalized Sturm-Liouville boundary value problems, but the present paper deals with a case that has not been studied before. To derive sampling theorems for the problem (1.1)-(1.7), we establish briefly some spectral properties and construct the Green’s function of the problem (1.1)-(1.7). Then we derive two sampling theorems using solutions and the Green’s function, respectively.
2 An operator formulation and asymptotic formulas
Some properties of the eigenvalues and asymptotic formulas for the eigenvalues and the corresponding eigenfunctions for the same problem were given in . We state the results briefly in this section.
These functions are entire in λ for all .
3 Green’s function
To study the completeness of the eigenvectors of A and hence the completeness of the eigenfunctions of the problem (1.1)-(1.7), we construct the resolvent of A as well as the Green’s function of the problem (1.1)-(1.7). We assume without any loss of generality that is not an eigenvalue of A.
4 The sampling theorems
and the fact that , and are entire functions of exponential type , we conclude that is also exponential type . □
The next theorem is devoted to giving interpolation sampling expansions associated with the problem (1.1)-(1.7) defined in terms of the Green’s function (these steps were introduced for the first time in ,  and recently in , ). As we see in (3.9), the Green’s function of the problem (1.1)-(1.7) has simple poles at . Let the function to be , where is a fixed point and is the function defined in (2.5) or the canonical product (4.14).
The series (4.16) converges absolutely on ℂ and uniformly on a compact subset of ℂ.
Combining (4.26), (4.27), and (4.18) under the assumption that for all n. If , for some n, the same expansion holds with . The convergence properties as well as the analytic and growth properties can be established as in Theorem 2. □
Now we give an example to illustrate the sampling transform.
where are the zeros of (4.32).
All authors are very grateful to the anonymous referees for their valuable suggestions.
- Paley R, Wiener N: The Fourier Transforms in the Complex Domain. Am. Math. Soc., Providence; 1934.Google Scholar
- Levinson N: Gap and Density Theorems. Am. Math. Soc., Providence; 1940.Google Scholar
- Kramer HP: A generalized sampling theorem. J. Math. Phys. 1959, 38: 68-72.View ArticleGoogle Scholar
- Zayed AI: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton; 1993.Google Scholar
- Everitt WN, Nasri-Roudsari G, Rehberg J: A note on the analytic form of the Kramer sampling theorem. Results Math. 1998, 34(3-4):310-319. 10.1007/BF03322057MathSciNetView ArticleGoogle Scholar
- Everitt WN, Nasri-Roudsari G: Interpolation and sampling theories, and linear ordinary boundary value problems. In Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Edited by: Higgins JR, Stens RL. Oxford University Press, Oxford; 1999. Chapter 5Google Scholar
- García AG, Littlejohn LL: On analytic sampling theory. J. Comput. Appl. Math. 2004, 171: 235-246. 10.1016/j.cam.2004.01.016MathSciNetView ArticleGoogle Scholar
- Everitt WN, García AG, Hernández-Medina MA: On Lagrange-type interpolation series and analytic Kramer kernels. Results Math. 2008, 51: 215-228. 10.1007/s00025-007-0271-3MathSciNetView ArticleGoogle Scholar
- Weiss P: Sampling theorems associated with Sturm Liouville system. Bull. Am. Math. Soc. 1957, 63: 242.Google Scholar
- Zayed AI, Hinsen G, Butzer PL: On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems. SIAM J. Appl. Math. 1990, 50: 893-909. 10.1137/0150053MathSciNetView ArticleGoogle Scholar
- Zayed AI: On Kramer’s sampling theorem associated with general Sturm-Liouville boundary value problems and Lagrange interpolation. SIAM J. Appl. Math. 1991, 51: 575-604. 10.1137/0151030MathSciNetView ArticleGoogle Scholar
- Everitt WN, Schöttler G, Butzer PL: Sturm-Liouville boundary value problems and Lagrange interpolation series. Rend. Mat. Appl. 1994, 14: 87-126.MathSciNetGoogle Scholar
- Annaby MH, Bustoz J, Ismail MEH: On sampling theory and basic Sturm-Liouville systems. J. Comput. Appl. Math. 2007, 206: 73-85. 10.1016/j.cam.2006.05.024MathSciNetView ArticleGoogle Scholar
- Boumenir A, Chanane B: Eigenvalues of S-L systems using sampling theory. Appl. Anal. 1996, 62: 323-334. 10.1080/00036819608840486MathSciNetView ArticleGoogle Scholar
- Boumenir A: The sampling method for SL problems with the eigenvalue in the boundary conditions. Numer. Funct. Anal. Optim. 2000, 21: 67-75. 10.1080/01630560008816940MathSciNetView ArticleGoogle Scholar
- Annaby MH, Tharwat MM: On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. SUT J. Math. 2006, 42(2):157-176.MathSciNetGoogle Scholar
- Annaby MH, Tharwat MM: On sampling and Dirac systems with eigenparameter in the boundary conditions. J. Appl. Math. Comput. 2011, 36(1-2):291-317. 10.1007/s12190-010-0404-9MathSciNetView ArticleGoogle Scholar
- Zayed AI, García AG: Kramer’s sampling theorem with discontinuous kernels. Results Math. 1998, 34: 197-206. 10.1007/BF03322050MathSciNetView ArticleGoogle Scholar
- Annaby MH, Freiling G, Zayed AI: Discontinuous boundary-value problems: expansion and sampling theorems. J. Integral Equ. Appl. 2004, 16(1):1-23. 10.1216/jiea/1181075255MathSciNetView ArticleGoogle Scholar
- Boumenir A, Zayed AI: Sampling with a string. J. Fourier Anal. Appl. 2002, 8(3):211-231. 10.1007/s00041-002-0009-2MathSciNetView ArticleGoogle Scholar
- Tharwat MM: Discontinuous Sturm Liouville problems and associated sampling theories. Abstr. Appl. Anal. 2011.Google Scholar
- Hıra F, Altınışık N: Eigenvalue problem with moving discontinuity points. J. Adv. Math. 2014, 9(2):2000-2010.Google Scholar
- Fulton CT: Two-point boundary value problems with eigenvalues parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A 1977, 77: 293-308. 10.1017/S030821050002521XMathSciNetView ArticleGoogle Scholar
- Altınışık N, Kadakal M, Mukhtarov OS: Eigenvalues and eigenfunctions of discontinuous Sturm Liouville problems with eigenparameter dependent boundary conditions. Acta Math. Hung. 2004, 102(1-2):159-175.Google Scholar
- Titchmarsh EC: Eigenfunctions Expansion Associated with Second Order Differential Equations. Part I. Oxford University Press, London; 1962.Google Scholar
- Levitan BM, Sargjan IS: Introduction to Spectral Theory Self-Adjoint Ordinary Differential Operators. Am. Math. Soc., Providence; 1975.Google Scholar
- Boas RP: Entire Functions. Academic Press, New York; 1954.Google Scholar
- Annaby MH, Zayed AI: On the use of Green’s function in sampling theory. J. Integral Equ. Appl. 1998, 10(2):117-139. 10.1216/jiea/1181074218MathSciNetView ArticleGoogle Scholar
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