Sampling theory for Sturm-Liouville problem with boundary and transmission conditions containing an eigenparameter. (English) Zbl 1339.34042
The well known Whittaker-Kotelnikov-Shannon sampling theorem has some important generalizations with different type of interpolations. Among them, the Kramer type interpolation uses a complete orthogonal set in \(L^2(I)\) with \(I\) being a closed interval. In this article, a Kramer-type sampling theorem associated with Sturm-Liouville problem which has two points of discontinuity and contains an eigenparameter in a boundary condition and also transmission conditions is considered and a Lagrange interpolation type reconstruction formula is given.
Reviewer: Chie-Ping Chu (Taipei)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 41A05 | Interpolation in approximation theory |
| 94A20 | Sampling theory in information and communication theory |
Keywords:
Whittaker-Shannon’s sampling theory; Kramer’s sampling theory; discontinuous Sturm-Liouville problemsReferences:
| [1] | Paley, R., Wiener, N.: Fourier transforms in the complex domain. Am. Math. Soc. Colloq. Publ. 19 (1934) · Zbl 0011.01601 |
| [2] | Zayed A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993) · Zbl 0868.94011 |
| [3] | Butzer, P. L.; Schmeisser, G. R.; Stens, L.; Marvasti, F. (ed.), An introduction to sampling analysis, 17-21 (2001), New York · doi:10.1007/978-1-4615-1229-5_2 |
| [4] | Levinson, N.: Gap and density theorems. Am. Math. Soc. Colloq. Publ. 26 (1940) · JFM 66.0332.01 |
| [5] | Kramer H.P.: A generalized sampling theorem. J. Math. Phys. 38, 68-72 (1959) · Zbl 0196.31702 |
| [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) · Zbl 0917.30024 · doi:10.1007/BF03322057 |
| [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) · Zbl 1136.94003 · doi:10.1007/s00025-007-0271-3 |
| [8] | García A.G., Littlejohn L.L.: On analytic sampling theory. J. Comput. Appl. Math. 171, 235-246 (2004) · Zbl 1057.44003 · doi:10.1016/j.cam.2004.01.016 |
| [9] | Everitt, W. N.; Nasri-Roudsari, G.; Higgins, J. R. (ed.); Stens, R. L. (ed.), Interpolation and sampling theories and linear ordinary boundary value problems (1999), Oxford |
| [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) · Zbl 0813.34028 |
| [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) · Zbl 0722.41008 · doi:10.1137/0151030 |
| [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) · Zbl 0695.41002 · doi:10.1137/0150053 |
| [13] | Boumenir A., Chanane B.: Eigenvalues of S-L systems using sampling theory. Appl. Anal. 62, 323-334 (1996) · Zbl 0864.34073 · doi:10.1080/00036819608840486 |
| [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) · Zbl 1128.39014 · doi:10.1016/j.cam.2006.05.024 |
| [15] | Boumenir A., Zayed A.I.: Sampling with a string. J. Fourier Anal. Appl. 8, 211-231 (2002) · Zbl 1046.94003 · doi:10.1007/s00041-002-0009-2 |
| [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) · Zbl 1133.34010 |
| [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) · Zbl 0948.34016 · doi:10.1080/01630560008816940 |
| [18] | Annaby M., Freiling G.: A sampling theorem for transforms with discontinuous kernels. Appl. Anal. 83, 1053-1075 (2004) · Zbl 1080.34065 · doi:10.1080/00036810410001657224 |
| [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) · Zbl 1089.34014 · doi:10.1216/jiea/1181075255 |
| [20] | Zayed A.I., García A.G.: Kramer’s sampling theorem with discontinuous kernels. Results Math. 34, 197-206 (1998) · Zbl 0914.41003 · doi:10.1007/BF03322050 |
| [21] | Kobayashi M.: Eigenfunction expansions: a discontinuous version. SIAM J. Appl. Math. 50, 910-917 (1990) · Zbl 0698.34020 · doi:10.1137/0150054 |
| [22] | Tharwat, M.M.: Discontinuous Sturm-Liouville problems and associated sampling theories. Abstr. Appl. Anal. doi:10.1155/2011/610232 (2011) · Zbl 1229.94039 |
| [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) · Zbl 1052.34030 · doi:10.1023/B:AMHU.0000023214.99631.52 |
| [24] | Kadakal M., Mukhtarov O.Sh.: Sturm Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367-1379 (2007) · Zbl 1140.34012 · doi:10.1016/j.camwa.2006.05.032 |
| [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) · Zbl 1044.34046 |
| [26] | Titchmarsh E.C.: Eigenfunctions Expansion Associated with Second Order Differential Equations I. Oxford University Press, London (1962) · Zbl 0099.05201 |
| [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) · Zbl 0302.47036 |
| [28] | Boas R.P.: Entire Functions. Academic Press, New York (1954) · Zbl 0058.30201 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
