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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Eigenvalues of periodic Sturm-Liouville problems by the Shannon-Whittaker sampling theorem


Author: Amin Boumenir
Journal: Math. Comp. 68 (1999), 1057-1066
MSC (1991): Primary 34L15, 42A15
Published electronically: February 10, 1999
MathSciNet review: 1627850
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Abstract | References | Similar Articles | Additional Information

Abstract: We are concerned with the computation of eigenvalues of a periodic Sturm-Liouville problem using interpolation techniques in Paley-Wiener spaces. We shall approximate the Hill discriminant by sampling a few of its values and then find its zeroes which are the square roots of the eigenvalues. Computable error estimates are provided together with eigenvalue enclosures.


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  • 1. Robert Dalmasso, Uniqueness of positive and nonnegative solutions of nonlinear equations, Funkcial. Ekvac. 37 (1994), no. 3, 461–482. MR 1311555 (97c:34037)
  • 2. Amin Boumenir, The transmutation operator for -𝑑²/(𝑤(𝑥)𝑑𝑥²), Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 1, 85–98. MR 1318624 (95m:34047), http://dx.doi.org/10.1017/S0308210500030766
  • 3. A. Boumenir and B. Chanane, Eigenvalues of S-L Systems using Sampling Theory, Applicable Analysis, Vol. 62, 1996, 323-334.
  • 4. M. S. P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, 1973.
  • 5. Charles T. Fulton and Steven A. Pruess, Erratum: “Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems” [J. Math.\ Anal.\ Appl.\ {188} (1994), no.\ 1, 297–340; MR1301734 (96e:34144)], J. Math. Anal. Appl. 189 (1995), no. 1, 313–314. MR 1312045 (97f:34071), http://dx.doi.org/10.1006/jmaa.1995.1019
  • 6. I. S. Kac and M. G. Krein, Spectral function of the string, Amer. Math. Soc. Transl. (2), 103, 1970, 19-103.
  • 7. B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088 (89b:34001)
  • 8. M. A. Naĭmark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. MR 0262880 (41 #7485)
  • 9. Frank Stenger, Approximations via Whittaker’s cardinal function, J. Approximation Theory 17 (1976), no. 3, 222–240. MR 0481786 (58 #1885)
  • 10. E. C. Titchmarsh, Eigenfunctions expansion, Part II, Oxford University Press, 1961.
  • 11. Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907 (95f:94008)

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Additional Information

Amin Boumenir
Affiliation: Department of Mathematics, College of Sciences, Sultan Qaboos University, P.O. Box 36, Alkhod 123, Muscat, Sultanate of Oman
Email: boumenir@squ.edu.om

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01053-4
PII: S 0025-5718(99)01053-4
Keywords: Periodic Sturm-Liouville problems, eigenvalue problem, interpolation, Shannon-Whittaker sampling theorem
Received by editor(s): June 9, 1997
Received by editor(s) in revised form: October 28, 1997
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society