ISSN:
1432-0940
Keywords:
33A65
;
34B20
;
Orthogonal polynomials
;
Singular differential equation
;
Legendre type boundary problem
;
Weighted Sobolev space
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$M_k [y] = \lambda y$$ 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 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01888171
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