The Sobolev orthogonality and spectral analysis of the Laguerre polynomials {Ln-k}for positive integers k

• Authors: W. N. Everitt;L. L. Littlejohn;R. Wellman
• Venue: Journal of Computational and Applied Mathematics - Special issue: On the occasion of the eightieth birthday of prof. W.M. Everitt
• Year: 2004

For k ∈ N, we consider the analysis of the classical Laguerre differential expression l-k[y](x)=1/x-ke-x(-(x-k+1e-xy'(x))'rx-ke-xy(x)) (x∈(0,∞)), where r≥ 0 is fixed, in several nonisomorphic Hilbert and Hilbert-Sobolev spaces. In one of these spaces, specifically the Hilbert space L2((0, ∞); x-ke-x), it is well known that the Glazman-Krein-Naimark theory produces a self-adjoint operator A-k, generated by l-k[ċ], that is bounded below by rl, where I is the identity operator on L2((0, ∞); x-ke-x). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {Hs,-k=(Vs,-k (ċ,ċ)s,-k)}s0 and left-definite self-adjoint operators {Bs,-k}s 0 associated with the pair L2((0, ∞); x-ke-x),A-k). For A-k and each of the operators Bs,-k, it is the case that the tail-end sequence {Ln-k}∞n=k of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces. In 1995, Kwon and Littlejohn introduced a Hilbert-Sobolev space Wk[0,∞) in which the entire sequence of Laguerre polynomials is orthonormal. In this paper, we construct a self-adjoint operator in this space, generated by the second-order Laguerre differential expression l-k[ċ], having {Ln-k}∞n=0as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of Wk[0, ∞) with the kth left-definite vector space Vk,-k.