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

Abstract

For k∈ N , we consider the analysis of the classical Laguerre differential expression ℓ −k[y](x)= 1 x −k e −x (−(x −k+1 e −xy′(x))′+rx −k e −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 L 2((0,∞); x − k e − x ), it is well known that the Glazman–Krein–Naimark theory produces a self-adjoint operator A − k , generated by ℓ − k [·], that is bounded below by rI, where I is the identity operator on L 2((0,∞); x − k e − x ). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces { H s,− k =( V s,− k ,(·,·) s,− k )} s>0 and left-definite self-adjoint operators { B s,− k } s>0 associated with the pair ( L 2((0,∞); x − k e − x ), A − k ). For A − k and each of the operators B s,− k , it is the case that the tail-end sequence { L n − 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 W k [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 [·], having { L n − k } n=0 ∞ as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of W k [0,∞) with the kth left-definite vector space V k,− k .