Constructive Approximation

, Volume 7, Issue 1, pp 485–500 | Cite as

The left-definite Legendre type boundary problem

  • W. N. Everitt
  • L. L. Littlejohn
  • S. C. Williams


The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM k [−] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation
$$M_k [y] = \lambda y$$
has orthogonal polynomial eigenfunctions, called the Legendre type polynomials, associated with the eigenvalues
$$\lambda _n = n(n + 1)(n^2 + n + 4\alpha - 2) + k.$$

In this paper, we show that the spaceC2[−1, 1] is dense inH, from which it follows that the spectrum of the self-adjoint left-definite operatorS k [·] associated withM k [·] is a purely point spectrum and consists only of the eigenvaluesλ n . Comparisons betweenS k [·] and the associated right-definite operatorT k [·] are made. This work extends earlier work of Everitt, Krall, Littlejohn, and Williams.

AMS classification

33A65 34B20 

Key words and phrases

Orthogonal polynomials Singular differential equation Legendre type boundary problem Weighted Sobolev space 


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Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • W. N. Everitt
    • 1
  • L. L. Littlejohn
    • 2
  • S. C. Williams
    • 2
  1. 1.Department of MathematicsUniversity of BirminghamBirminghamEngland
  2. 2.Department of MathematicsUtah State UniversityLoganUSA

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