On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

摘要

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L~2(-1, 1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L~2(-1, 1), A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A_1, first observed by Everitt, and the self-adjoint operator A~(1/2). In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Maric result which gives optimal global smoothness of functions in D(A).