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http://dx.doi.org/10.4134/JKMS.j150226

SAMPLING THEOREMS ASSOCIATED WITH DIFFERENTIAL OPERATORS WITH FINITE RANK PERTURBATIONS  

Annaby, Mahmoud H. (Department of Mathematics Faculty of Science Cairo University)
El-Haddad, Omar H. (Department of Mathematics Faculty of Science Cairo University)
Hassan, Hassan A. (Department of Mathematics Faculty of Science Cairo University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 969-990 More about this Journal
Abstract
We derive a sampling theorem associated with first order self-adjoint eigenvalue problem with a finite rank perturbation. The class of the sampled integral transforms is of finite Fourier type where the kernel has an additional perturbation.
Keywords
Green's function; eigenfunctions expansions; finite rank perturbations;
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1 M. H. Annaby, On sampling theory associated with the resolvents of singular Sturm-Liouville problems, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1803-1812.   DOI
2 M. H. Annaby, G. Freiling, and A. I. Zayed, Discontinuous boundary-value problems:Expansion and sampling theorems, J. Integral Equations Appl. 16 (2004), no. 1, 1-23.   DOI
3 M. H. Annaby and H. A. Hassan, A sampling theorem associated with boundary-value problems with not necessarily simple eigenvalues, Int. J. Math. Math. Sci. 21 (1998), no. 3, 571-580.   DOI
4 M. H. Annaby, H. A. Hassan, and O. H. El-Haddad, Perturbed discrete Sturm-Liouville problems and associated sampling theorems, Rocky Mountain J. Math. 39 (2009), no. 6, 1781-1807.   DOI
5 M. H. Annaby, H. A. Hassan, and O. H. El-Haddad, A perturbed Whittaker-Kotel'nikov-Shannon sampling theorem, J. Math. Anal. Appl. 381 (2011), no. 1, 64-79.   DOI
6 M. H. Annaby and A. I. Zayed, On the use of Green's function in sampling theory, J. Integral Equations Appl. 10 (1998), no. 2, 117-139.   DOI
7 R. Boas, Entire Functions, Academic Press, New York, 1954.
8 P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birhkhauser, Basel, 1971.
9 P. L. Butzer, G. Schmeisser, and R. L. Stens, An introduction to sampling analysis, Nonuniform sampling, 17-121, Inf. Technol. Transm. Process. Storage, Kluwer/Plenum, New York, 2001.
10 P. L. Butzer and G. Schottler, Sampling theorems associated with fourth and higher order self-adjoint eigenvalue problems, J. Comput. Appl. Math. 51 (1994), no. 2, 159-177.   DOI
11 L. L. Campbell, A comparison of the sampling theorems of Kramer and Whittaker, SIAM J. Appl. Math. 12 (1964), 117-130.   DOI
12 E. A. Catchpole, A Cauchy problem for an ordinary integro-differential equation, Proc. Roy. Soc. Edinburgh Sect. A 72 (1974), no. 1, 39-55.
13 J. A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, 1972.
14 W. N. Everitt and G. Nasri-Roudsari, Sturm-Liouville problems with coupled boundary conditions and Lagrange interpolation series, J. Comput. Anal. Appl. 1 (1999), no. 4, 319-347.
15 W. N. Everitt and G. Nasri-Roudsari, Sturm-Liouville problems with coupled boundary conditions and Lagrange in-terpolation series II, Rend. Mat. Appl. (7) 20 (2000), 199-238.
16 J. R. Higgins, Sampling Theorey in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, 1996.
17 W. N. Everitt and A. Poulkou, Kramer analytic kernels and first-order boundary value problems, J. Comput. Appl. Math. 148 (2002), no. 1, 22-47.
18 I. Gohberg and S. Goldberg, Basic Operator Theory, Birkhauser, Boston, 1980.
19 A. H. Haddad, K. Yao, and J. B. Thomas, General methods for the derivation of sam-pling theorems, IEEE Trans. Inform. Theory 13 (1967), 227-230.   DOI
20 V. Kotel'nikov, On the carrying capacity of the ether and wire in telecommunications, (Russian) Material for the first all union conference on questions of communications, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933.
21 M. A. Naimark, Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators, George Harrap, London, 1967.
22 R. Paley and N. Wiener Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloquium Publ. Ser. Vol 19, Amer. Math. Soc., Providence, RI, 1934.
23 C. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10-21.
24 L. O. Silva and J. H. Toloza, Bounded rank-one perturbations in sampling theory, J. Math. Anal. Appl. 345 (2008), no. 2, 661-669.   DOI
25 I. Stakgold, Green's Functions and Boundary Value Problems, John Wiley, New York, 1987.
26 E. Whittaker, On the functions which are represented by the expansion of the interpo-lation theory, Proc. Roy. Soc. Edinburgh Sec. A 35 (1915), 181-194.   DOI
27 A. I. Zayed, Advances in Shannon's Sampling Theory, CRC, Boca Raton, 1993.