One and multidimensional sampling theorems associated with Dirichlet problems

Mathematical Methods in the Applied Sciences

Volumn 21, Issue 4, 1998, Pages 361-374

  Annaby, Mahmoud H ^a  

^a RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; ERROR ANALYSIS; INTEGRAL EQUATIONS; INTERPOLATION; LAGRANGE MULTIPLIERS; MATHEMATICAL TRANSFORMATIONS; THEOREM PROVING;

DIRICHLET PROBLEMS; SAMPLING THEOREMS;

GREEN'S FUNCTION;

EID: 0032022910     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1476(19980310)21:4<361::AID-MMA954>3.0.CO;2-E     Document Type: Article

References (32)

1. Annaby, M. H. and Zayed A. I., 'On the use of Green's function in sampling theory', submitted.
2. Annaby, M. H. and El-Saved, M. A., 'Kramer-type sampling theorems associated with Fredholm integral operators', Meth. Appl. Anal., 2, 76-91 (1995).
3. Boas, R., Entire functions, Academic Press, New York, 1954.
4. Butzer, P. L., 'A survey of the Whittaker-Shannon sampling theorem and some of its extensions', J. Math. Res. Exposition, 3, 185-212 (1983).
5. Butter, P. L. and Hinsen, G. 'Two dimensional non uniform sampling expansions - an iterative approach, I: theory of two dimensional band-limited signals', Appl. Anal., 32, 53-67 (1989); II: Reconstruction formulae and applications. Appl. Anal., 32, 69-85 (1989).
6. Butter, P. L. and Hinsen, G. 'Two dimensional non uniform sampling expansions - an iterative approach, I: theory of two dimensional band-limited signals', Appl. Anal., 32, 53-67 (1989); II: Reconstruction formulae and applications. Appl. Anal., 32, 69-85 (1989).
7. Bulzer, P. L. and Nasri-Roudsrai, G., 'Kramer's sampling theorem in signal analysis and its role in Mathematics', in: Image Proceeding; Mathematical Methods and Applications I.M.A. Conf. Proc. 1994, pp. 49-95, Clarendon Press, Oxford, 1997.
8. Butzer, P. L. and Schöttler, G., 'Sampling theorems associated with fourth and higher order self-adjoint eigenvalue problems', J. Comput. Appl. Math., 51, 159-171 (1994).
9. Butzer, P. L. and Splettstösser, W., 'A sampling theorem for duration-limited functions with error estimates', Inform. Control, 34, 55-65 (1977).
10. Cochran, J. A., The Analysis, of Linear Integral Equations, McGraw-Hill, New York, 1972.
11. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
12. Gohberg, I and Goldberg, S., Basic Operator Theory, Birkhäuser, Boston, 1981.
13. Haddad, A., Yao, K. and Thomas, J., 'General methods for the derivation of the sampling theorems', IEEE Trans. Inform. Theory, IT-13, 327-230 (1967).
14. Higgins, J. R., 'Five short stories about the cardinal series', Bull. Amer. Math. Soc., 12, 45-89 (1989).
15. Higgins, J. R., Sampling Theory in Fourier and Signal Analysis Foundations, Oxford University Press, Oxford, 1996.
16. Jerry, A. J., 'The Shannon sampling theorem-its various extensions and applications. A tutorial review', Proc. IEEE, 65, 1565-1598 (1977).
17. Jerry, A. J., 'Error analysis in application of generalizations of the sampling theorem', in: Advanced Topics in Shannon Sampling and Interpolation Theory, pp. 219-298. Springer, New York, 1993.
18. Kramer, H. P., 'A generalized sampling theorem', J. Math. Phys., 38, 68-72 (1959).
19. Levinson, N., Gap and Density Theorems, Amer Math. Soc. Providence, RI, 1940.
20. Paley, R. and Wiener, N., Fourier Transforms in the Complex Domain, Amer Math Soc. Colloq, Publ., 19, 1934.
21. Petrovski, I. G., Lectures on The Theory of Integral Equations, Mir Publisher, Moscow, 1971.
22. Roach, G. F., Green's Function Introductory Theory with Applications, VNR, London. 1970.
23. Shannon, C. E., 'Communications in the presence of noise', Proc 1RE, 37, 10-21 (1949).
24. Splettstosser, W., 'Error estimates for sampling approximation of non-bandlimited functions', Math. Meth. Appl. Sci. 1, 127-137 (1979).
25. Titchmarsh, E., Eigenfunction Expansions Associated with Second Order Differential Equations Part II, Clarendon Press, Oxford, 1958.
26. Titchmarsh, E., Eigenfunction Expansions Associated with Second Order Differential Equations Part I, Clarendon Press, Oxford, 1962.
27. Weiss, P., 'Sampling theorems associated with Sturm-Liouville systems', Bull. Amer. Math. Soc. 163, 242 (1957)
28. Whittaker, E., 'On the functions which are represented by the expansion of the interpolation theory', Proc. Roy. Soc. Edin. Sect. A, 35, 181-194 (1915).
29. Zayed, A. I., 'Kramer's sampling theorem for multidimensional Signals and its relationship with Lagrange-type interpolation', J. Multidimensional Systems Signal Process, 3, 323-340 (1992).
30. Zayed, A. I., 'A new role of Green's function in interpolation and sampling theory', J. Math. Anal. Appl, 175, 222-238 (1993).
31. Zayed, A. I., Adrances on Shannon's Sampling Theory, CRC Press. Boca Raton, 1993.
32. Zayed, A. I., El-Sayed, M. A. and Annaby M. H., 'On Lagrange interpolations and Kramer's sampling theorem associated with self-adjoing boundary-value problems'. J. Math. Anal. Appl., 158, 269-284 (1991).