Abstract
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
has orthogonal polynomial eigenfunctions, called the Legendre type polynomials, associated with the eigenvalues
In this paper, we show that the spaceC 2[−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.
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Communicated by Mourad Ismail.
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Everitt, W.N., Littlejohn, L.L. & Williams, S.C. The left-definite Legendre type boundary problem. Constr. Approx 7, 485–500 (1991). https://doi.org/10.1007/BF01888171
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DOI: https://doi.org/10.1007/BF01888171