Littlejohn, Lance, 1951-Tuncer, Davut.Baylor University. Dept. of Mathematics.2010-02-022010-02-022009-122010-02-02http://hdl.handle.net/2104/5538Littlejohn and Wellman developed a general abstract left-definite theory for a self-adjoint operator A that is bounded below in a Hilbert space (H,(‧,‧)). More specifically, they construct a continuum of Hilbert spaces {(H_r,(‧,‧)_r)}_r>0 and, for each r>0, a self-adjoint restriction A_r of A in H_r. The Hilbert space H_r is called the rth left-definite Hilbert space associated with the pair (H,A) and the operator A_r is called the rth left-definite operator associated with (H,A). We apply this left-definite theory to the self-adjoint Legendre type differential operator generated by the fourth-order formally symmetric Legendre type differential expression ℓ[y](x):=((1-x²)²y″(x))″-((8+4A(1-x²))y′(x))′ +λy(x), where the numbers A and λ are, respectively, fixed positive and non-negative parameters and where x ∈ (-1,1).197468861 bytes707496 bytesapplication/pdfapplication/pdfenBaylor University theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact librarywebmaster@baylor.edu for inquiries about permission.Legendre Type Differential Equation.Self-adjoint Operator Theory.Spectral Analysis.Orthogonal Polynomials.Special Functions.Legendre Type Orthogonal Polynomials.Left-Definite Theory.Combinotorics.ThesisWorldwide access.Access changed 3/18/13.