{"id":{"repo_id":"baylor","oai_identifier":"oai:baylor-ir.tdl.org:2104/5327"},"canonical_url":"https://search.dev.ndltd.org/etd/baylor/oai:baylor-ir.tdl.org:2104/5327","repository":{"repo_id":"baylor","name":"Baylor University","base_url":"https://baylor-ir.tdl.org/server/oai/request"},"display":{"title":"Applied left-definite theory : the Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators.","abstract":"It is well known that, for –α, –β, –α – β – 1 ∉ ℕ, the Jacobi polynomials {Pn(α,β)(x)} ∞ n=0 are orthogonal on ℝ with respect to a bilinear form of the type(f,g)μ = ∫ℝfgdμ, for some measure μ. However, for negative integer parameters α and β, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to a bilinear form of this type for any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where α = β = –1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results will then be generalized to the case where α > –1, β = –1.","abstract_html":"It is well known that, for –α, –β, –α – β – 1 ∉ ℕ, the Jacobi polynomials {Pn(α,β)(x)} ∞ n=0 are orthogonal on ℝ with respect to a bilinear form of the type(f,g)μ = ∫ℝfgdμ, for some measure μ. However, for negative integer parameters α and β, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to a bilinear form of this type for any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where α = β = –1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results will then be generalized to the case where α > –1, β = –1.","abstract_has_math":false,"creators":["Bruder, Andrea S."],"institution":"Baylor University.","degree_name":"Ph.D.","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Littlejohn, Lance, 1951-"],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-06","date_published":"2009-06","updated_at":"2026-07-24T01:07:52Z","subjects":["Jacobi polynomials.","Sobolev spaces.","Selfadjoint operators.","Eigenfunctions."],"languages":["en"],"rights":["Baylor University works 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 libraryquestions@baylor.edu for inquiries about permission."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2104/5327","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Littlejohn, Lance, 1951-"]},{"key":"dc:creator","label":"Author","values":["Bruder, Andrea S."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2009-06-10T12:58:30Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2009-06-10T12:58:30Z"]},{"key":"dc:date.issued","label":"Date","values":["2009-06"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Baylor University."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Jacobi polynomials.","Sobolev spaces.","Selfadjoint operators.","Eigenfunctions."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Baylor University works 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 libraryquestions@baylor.edu for inquiries about permission."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/2104/5327"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It is well known that, for –α, –β, –α – β – 1 ∉ ℕ, the Jacobi polynomials {Pn(α,β)(x)} ∞ n=0 are orthogonal on ℝ with respect to a bilinear form of the type(f,g)μ = ∫ℝfgdμ, for some measure μ. However, for negative integer parameters α and β, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to a bilinear form of this type for any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where α = β = –1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results will then be generalized to the case where α > –1, β = –1."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Applied left-definite theory : the Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators."]}]}],"canonical_facts":{"dc:contributor.advisor":["Littlejohn, Lance, 1951-"],"dc:creator":["Bruder, Andrea S."],"dc:date.accessioned":["2009-06-10T12:58:30Z"],"dc:date.available":["2009-06-10T12:58:30Z"],"dc:date.issued":["2009-06"],"dc:description.abstract":["It is well known that, for –α, –β, –α – β – 1 ∉ ℕ, the Jacobi polynomials {Pn(α,β)(x)} ∞ n=0 are orthogonal on ℝ with respect to a bilinear form of the type(f,g)μ = ∫ℝfgdμ, for some measure μ. However, for negative integer parameters α and β, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to a bilinear form of this type for any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where α = β = –1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results will then be generalized to the case where α > –1, β = –1."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/2104/5327"],"dc:language.iso":["en"],"dc:rights":["Baylor University works 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 libraryquestions@baylor.edu for inquiries about permission."],"dc:subject":["Jacobi polynomials.","Sobolev spaces.","Selfadjoint operators.","Eigenfunctions."],"dc:title":["Applied left-definite theory : the Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators."],"dc:type":["Thesis"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["Baylor University."]},"updated_at":"2026-07-24T01:07:52Z"}