{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/37448"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/37448","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A duality approach to spline approximation","abstract":"This dissertation discusses a new approach to spline approximation. A periodic spline approximation 𝑓<sub>M,m,N</sub>(x) = Σ<sub>k=1</sub><sup>N</sup>α<sub>k</sub>Φ<sub>M,k</sub>(x) to a periodic function 𝑓(x) is determined by requiring < Φ<sub>m,j</sub>, 𝑓 - 𝑓<sub>M,m,N</sub> > = 0 for j = 1,...,N, where the Φ<sub>L,k</sub>'s are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored.","abstract_html":"This dissertation discusses a new approach to spline approximation. A periodic spline approximation 𝑓&lt;sub&gt;M,m,N&lt;/sub&gt;(x) = Σ&lt;sub&gt;k=1&lt;/sub&gt;&lt;sup&gt;N&lt;/sup&gt;α&lt;sub&gt;k&lt;/sub&gt;Φ&lt;sub&gt;M,k&lt;/sub&gt;(x) to a periodic function 𝑓(x) is determined by requiring &lt; Φ&lt;sub&gt;m,j&lt;/sub&gt;, 𝑓 - 𝑓&lt;sub&gt;M,m,N&lt;/sub&gt; &gt; = 0 for j = 1,...,N, where the Φ&lt;sub&gt;L,k&lt;/sub&gt;&#x27;s are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored.","abstract_has_math":false,"creators":["Bonawitz, Elizabeth Ann"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Russell, David L."],"committee_members":["Johnson, Lee W.","Rogers, Robert C.","Kohler, Werner E.","Sun, Shu-Ming"],"year":1994,"date_issued":"1994-04-21","date_published":"1994-04-21","updated_at":"2026-07-22T22:18:46Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-03022006-093404"],"render_values":[{"text":"etd-03022006-093404","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/37448","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Russell, David L."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Johnson, Lee W.","Rogers, Robert C.","Kohler, Werner E.","Sun, Shu-Ming"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Bonawitz, Elizabeth Ann"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:09:54Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:09:54Z","2006-03-02"]},{"key":"dc:date.issued","label":"Date","values":["1994-04-21"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"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":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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A periodic spline approximation 𝑓<sub>M,m,N</sub>(x) = Σ<sub>k=1</sub><sup>N</sup>α<sub>k</sub>Φ<sub>M,k</sub>(x) to a periodic function 𝑓(x) is determined by requiring < Φ<sub>m,j</sub>, 𝑓 - 𝑓<sub>M,m,N</sub> > = 0 for j = 1,...,N, where the Φ<sub>L,k</sub>'s are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A duality approach to spline approximation"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Russell, David L."],"dc:contributor.committeemember":["Johnson, Lee W.","Rogers, Robert C.","Kohler, Werner E.","Sun, Shu-Ming"],"dc:contributor.department":["Mathematics"],"dc:creator":["Bonawitz, Elizabeth Ann"],"dc:date.accessioned":["2014-03-14T21:09:54Z"],"dc:date.available":["2014-03-14T21:09:54Z","2006-03-02"],"dc:date.issued":["1994-04-21"],"dc:description.abstract":["This dissertation discusses a new approach to spline approximation. 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