On the right-definite and left-definite theory of the Legendre polynomials

Abstract

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in $L (-1,1)$, 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 $(L (-1,1),A)$. 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-Mari result which gives optimal global smoothness of functions in D(A).