{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/106206"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/106206","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Notes on generalized Fourier series with application to gravitational field determination","abstract":"Let{(}φ<sub>n</sub>(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φ<sub>n</sub>(x)} is the series ∑<sub>n=0</sub><sup>∞</sup> (f, φ<sub>n</sub>) φ<sub>n</sub>(x), where (f, φ<sub>n</sub>) is the inner product of the functions f an φ<sub>n</sub>. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems.","abstract_html":"Let{(}φ&lt;sub&gt;n&lt;/sub&gt;(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φ&lt;sub&gt;n&lt;/sub&gt;(x)} is the series ∑&lt;sub&gt;n=0&lt;/sub&gt;&lt;sup&gt;∞&lt;/sup&gt; (f, φ&lt;sub&gt;n&lt;/sub&gt;) φ&lt;sub&gt;n&lt;/sub&gt;(x), where (f, φ&lt;sub&gt;n&lt;/sub&gt;) is the inner product of the functions f an φ&lt;sub&gt;n&lt;/sub&gt;. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems.","abstract_has_math":false,"creators":["Blackshear, Walter Thomas"],"institution":"Virginia Polytechnic Institute","degree_name":"M.S.","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1967,"date_issued":"1967","date_published":"1967","updated_at":"2026-07-22T22:20:26Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/106206","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Blackshear, Walter Thomas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2021-10-26T20:10:37Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2021-10-26T20:10:37Z"]},{"key":"dc:date.issued","label":"Date","values":["1967"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/106206"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let{(}φ<sub>n</sub>(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φ<sub>n</sub>(x)} is the series ∑<sub>n=0</sub><sup>∞</sup> (f, φ<sub>n</sub>) φ<sub>n</sub>(x), where (f, φ<sub>n</sub>) is the inner product of the functions f an φ<sub>n</sub>. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["M.S."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Notes on generalized Fourier series with application to gravitational field determination"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Blackshear, Walter Thomas"],"dc:date.accessioned":["2021-10-26T20:10:37Z"],"dc:date.available":["2021-10-26T20:10:37Z"],"dc:date.issued":["1967"],"dc:description.abstract":["Let{(}φ<sub>n</sub>(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φ<sub>n</sub>(x)} is the series ∑<sub>n=0</sub><sup>∞</sup> (f, φ<sub>n</sub>) φ<sub>n</sub>(x), where (f, φ<sub>n</sub>) is the inner product of the functions f an φ<sub>n</sub>. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems."],"dc:description.degree":["M.S."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/106206"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Notes on generalized Fourier series with application to gravitational field determination"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["M.S."],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:20:26Z"}