Hinton and Shaw’s Extension with Two Singular Points

  • Allan M. Krall
Part of the Operator Theory: Advances and Applications book series (OT, volume 133)


This chapter extends the results of the previous one to cover the situation that occurs when both a and b are singular points. The technique is similar. We restrict our attention to an interval (a′, b′) within (a, b), develop two M(λ) functions, one for generating L A 2 solutions near a, one for generating L A 2 solutions near b, by letting a′ → a, b′ → b.


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  1. [1]
    W. N. Everitt, Integrable-square solutions of ordinary differential equations, Quar. J. Math., Oxford (2) 10 (1959), 145–155.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D. B. Hinton and J. K. Shaw, Titchmarsh’s λ-dependent boundary conditions for Hamiltonian systems, Lect. Notes in Math. (Springer-Verlag), 964 (1982), 318–326.CrossRefGoogle Scholar
  3. [3]
    J. K. Shaw —, Hamiltonian systems of limit point or limit circle type with both ends singular, J. Diff. Eq. 50 (1983), 444–464.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    J. K. Shaw —, On boundary value problems for Hamiltonian systems with two singular points, SIAM J. Math. Anal. 15 (1984), 272–286.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edin. 87 (1981), 271–288.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. M. Krall, B. D. Hinton and J. K. Shaw, Boundary conditions for differential systems in intermediate limit situations, Proc. Conf. on Ordinary and Partial Differential Equations, ed. I. W. Knowles and R. E. Lewis, North Holland, 1984, 301–305.Google Scholar
  7. [7]
    L. L. Littlejohn and A. M. Krall, Orthogonal polynomials and singular Sturm-Liouville systems, I Rocky Mt. J. Math., 1987.Google Scholar
  8. [8]
    A. M. Krall —, Orthogonal polynomials and higher order singular Sturm-Liouville systems, Acta Applicandae Mathematical, (1989), 99–170.Google Scholar
  9. [9]
    E. C. Titchmarsh, Eigenfunction Expansions Oxford Univ. Press, Oxford, 1962.zbMATHGoogle Scholar

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© Springer Basel AG 2002

Authors and Affiliations

  • Allan M. Krall
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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