On sampling theory associated with the resolvents of singular Sturm-Liouville problems

Annaby, M.

doi:10.1090/S0002-9939-02-06727-8

On sampling theory associated with the resolvents of singular Sturm-Liouville problems
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by M. H. Annaby PDF

Proc. Amer. Math. Soc. 131 (2003), 1803-1812 Request permission

Abstract:

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green’s functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

References

K. Balachandran and S. Ilamaran, On nonlinear integrodifferential equations with deviating arguments, Tamkang J. Math. 26 (1995), no. 2, 149–154. MR 1356190, DOI 10.5556/j.tkjm.26.1995.4391
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. MR 1646834, DOI 10.1216/jiea/1181074218
M. H. Annaby and G. Freiling, Sampling expansions associated with Kamke problems, Math. Z. 234 (2000), no. 1, 163–189. MR 1759496, DOI 10.1007/s002090050508
M.H. Annaby and P.L. Butzer, On sampling associated with singular Sturm-Liouville eigenvalue problems: the limit-circle case, to appear.
P.L. Butzer and G. Nasri-Roudsari, Kramer’s sampling theorem in signal analysis and its role in mathematics. In: Image Processing, Mathematical Methods and Applications, The Institute of Mathematics and its Applications, New series, No 61, Clarendon Press, Oxford 1997, 49–95.
James Alan Cochran, The analysis of linear integral equations, McGraw-Hill Series in Modern Applied Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1972. MR 0447991
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
W. N. Everitt, G. Schöttler, and P. L. Butzer, Sturm-Liouville boundary value problems and Lagrange interpolation series, Rend. Mat. Appl. (7) 14 (1994), no. 1, 87–126 (English, with English and Italian summaries). MR 1284905
W. N. Everitt, A note on the self-adjoint domains of second-order differential equations, Quart. J. Math. Oxford Ser. (2) 14 (1963), 41–45. MR 143986, DOI 10.1093/qmath/14.1.41
C.T. Fulton, Parametrizations of Titchmarsh’s ’$m(\lambda )$’-Function in The Limit-Circle Case, Doctoral thesis, RWTH-Aachen, 1973.
Charles T. Fulton, Parametrizations of Titchmarsh’s $m(\lambda )$-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51–63. MR 450657, DOI 10.1090/S0002-9947-1977-0450657-6
Charles T. Fulton, Expansions in Legendre functions, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 130, 215–222. MR 657125, DOI 10.1093/qmath/33.2.215
A. Haddad, K. Yao and J. Thomas, General methods for the derivation of the sampling theorems, IEEE Trans. Inf. Theory, IT-13 (1967), 227–230.
H. P. Kramer, A generalized sampling theorem, J. Math. and Phys. 38 (1959/60), 68–72. MR 103786, DOI 10.1002/sapm195938168
N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795, DOI 10.1063/1.3047047
M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967. MR 0216050
Ivar Stakgold, Green’s functions and boundary value problems, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1487078
E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
Ahmed I. Zayed, Guido Hinsen, and Paul L. Butzer, On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems, SIAM J. Appl. Math. 50 (1990), no. 3, 893–909. MR 1050920, DOI 10.1137/0150053
Ahmed I. Zayed, On Kramer’s sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation, SIAM J. Appl. Math. 51 (1991), no. 2, 575–604. MR 1095036, DOI 10.1137/0151030
Ahmed I. Zayed, A new role of Green’s function in interpolation and sampling theory, J. Math. Anal. Appl. 175 (1993), no. 1, 222–238. MR 1216757, DOI 10.1006/jmaa.1993.1164
Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907
Ahmed I. Zayed, Sampling in a Hilbert space, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3767–3776. MR 1343731, DOI 10.1090/S0002-9939-96-03526-5