On sampling associated with singular sturm-liouville eigenvalue problems: The limit-circle case

Rocky Mountain Journal of Mathematics

Volumn 32, Issue 2, 2002, Pages 443-466

  Annaby, M H ^a   Butzer, P L ^b  

^a CAIRO UNIVERSITY   (Egypt)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Bessel s functions; Legendre functions of first and second kind; Sampling theory; Singular Sturm Liouville problems

Indexed keywords

EID: 0036624781     PISSN: 00357596     EISSN: None     Source Type: Journal    
DOI: 10.1216/rmjm/1030539680     Document Type: Article

References (26)

1. F.V. Atkinson and C.T. Fulton, Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit circle singularity, Proc. Royal Soc. Edinburgh 99 (1984), 51-70.
2. F.V. Atkinson and C.T. Fulton, Asymptotic formulas for eigenvalues of limit circle problems on a half line, Ann. Math. Pura Appl. 135 (1984), 363-398.
3. F.V. Atkinson and C.T. Fulton, Some limit circle eigenvalue problems and asymptotic formulae for eigenvalues, Lecture Notes Math. 964 (1982), 25-55.
4. P.L. Butzer and M. Hauss, Applications of sampling theory to combinatorial analysis, Stirling numbers, special functions and the Riemann zeta function, in Sampling theory in Fourier and signal analysis: Advanced topics, J.R. Higgins and R.L. Stens, eds., Oxford University Press, Oxford, (1999), 1-37.
5. 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, Clarendon Press, Oxford, (1997), 49-95.
6. P.L Butzer and G. Schottler, Sampling expansions for Jacobi functions and their applications, Acta Sci. Math. 57 (1993), 305-327.
7. P.L. Butzer and R. Stens and M. Wehrens, The continuous Legendre transform, its inverse transform, and applications, Internat. J. Math. Math. Sci. 3 (1980), 47-67.
8. E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw Hill, New York, 1955.
9. W.N. Everitt, Legendre polynomials and singular differential operators, Lecture Notes in Math. 827 (1980), 83-106.
10. W.N. Everitt, A note on the self-adjoint domains of second-order differential equations, Quart. J. Math. 13 (1963), 41-45.
11. W.N. Everitt, G. Schottler and P.L. Butzer, Sturm-Liouville boundary value problems and Lagrange interpolation series, Rend. Math. Appl. 14 (1994), 87-126.
12. m(X) ’-functions in the limit-circle cases, Doctoral thesis, RWTH-Aachen, 1973.
13. C.T. Fulton, Parametrizations of Titchmarsh’s m(X)-functions in the limit-circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63.
14. C.T. Fulton, Expansions in Legendre functions, Quart. J. Math. 33 (1982), 215-222.
15. I. Gradshleyn and I. Ryzhik, Tables of integrals, series and products, Academic Press, New York, 1965.
16. M. Hajmirzaahmad and A.M. Krall, Singular second order operators: The maximal and minimal operators and self-adjoint operator in between, SIAM Review 34 (1994), 614-634.
17. P. Hartman, Ordinary differential equations, John Wiley & Sons, New York, 1964.
18. E. Hille, Some problems concerning spherical harmonics, Arkiv. Mat. 13 (1918), 1-76.
19. E.W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge University Press, London, 1931.
20. N.N. Lebedev, Special functions and their applications, Prentice-Hall, Englewood Cliffs, N.J., 1965.
21. M.A. Naimark, Linear differential operators, Part II, George Harrap, London, 1967.
22. G. Sansone, Orthogonal functions, Interscience Publishers, Inc., New York, 1959.
23. N.M. Temme, Special functions, An introduction to the classical functions of mathematical physics, John Wiley, New York, 1996.
24. E.C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, Part I, Clarendon Press, Oxford, 1962.
25. A.I. Zayed, On Kramer’s sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation, SIAM J. Appl. Math. 51 (1991), 575-604.
26. A.I. Zayed and G.G. Walter, On the inversion of integral transforms associated with Sturm-Liouville problems, J. Math. Anal. Appl. 164 (1992), 285-306.