{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/44398"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/44398","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Error term improvements for van der Corput transforms","abstract":"Made available in DSpace on 2013-05-24T22:10:16Z (GMT). No. of bitstreams: 3 Joseph_Vandehey.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5) Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) license.txt: 4065 bytes, checksum: 629f12a987f803dfc9a1edc8c6d1c34c (MD5)","abstract_html":"Made available in DSpace on 2013-05-24T22:10:16Z (GMT). No. of bitstreams: 3 Joseph_Vandehey.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5) Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) license.txt: 4065 bytes, checksum: 629f12a987f803dfc9a1edc8c6d1c34c (MD5)","abstract_has_math":false,"creators":["Vandehey, Joseph"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ford, Kevin","Boca, Florin","Athreya, Jayadev S.","Hildebrand, A.J.","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-24T22:10:16Z","date_published":"2013-05-24T22:10:16Z","updated_at":"2026-07-22T22:25:34Z","subjects":["Asymptotic analysis","exponential sum","trigonometric sum","van der Corput transform"],"languages":["en"],"rights":["Copyright 2013 Joseph Vandehey"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/44398","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ford, Kevin","Boca, Florin","Athreya, Jayadev S.","Hildebrand, A.J.","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Vandehey, Joseph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-24T22:10:16Z","2013-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Asymptotic analysis","exponential sum","trigonometric sum","van der Corput transform"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Joseph Vandehey"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/44398"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Made available in DSpace on 2013-05-24T22:10:16Z (GMT). No. of bitstreams: 3 Joseph_Vandehey.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5) Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) license.txt: 4065 bytes, checksum: 629f12a987f803dfc9a1edc8c6d1c34c (MD5)","We improve the error term in the van der Corput transform for exponential sums, \\sum g(n) exp(2 \\pi i f(n)). For many smooth functions g and f, we can show that the largest factor of the error term is given by a simple explicit function, which can be used to show that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a, b] with far better results. As an example of the strength of these results, we provide a detailed analysis of the error term in the case g(x) = 1 and f(x) = (x/3)^(3/2).","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-04-16T13:54:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) Vandehey_Joseph.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5)"]},{"key":"dc:title","label":"Title","values":["Error term improvements for van der Corput transforms"]}]}],"canonical_facts":{"dc:contributor":["Ford, Kevin","Boca, Florin","Athreya, Jayadev S.","Hildebrand, A.J.","Zaharescu, Alexandru"],"dc:creator":["Vandehey, Joseph"],"dc:date":["2013-05-24T22:10:16Z","2013-05"],"dc:description":["Made available in DSpace on 2013-05-24T22:10:16Z (GMT). No. of bitstreams: 3 Joseph_Vandehey.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5) Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) license.txt: 4065 bytes, checksum: 629f12a987f803dfc9a1edc8c6d1c34c (MD5)","We improve the error term in the van der Corput transform for exponential sums, \\sum g(n) exp(2 \\pi i f(n)). For many smooth functions g and f, we can show that the largest factor of the error term is given by a simple explicit function, which can be used to show that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a, b] with far better results. As an example of the strength of these results, we provide a detailed analysis of the error term in the case g(x) = 1 and f(x) = (x/3)^(3/2).","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-04-16T13:54:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Vandehey_Joseph.tex: 217871 bytes, checksum: 782d147a858ee8301c78301b0ab1e4bd (MD5) Vandehey_Joseph.pdf: 931014 bytes, checksum: 406769988507aa24c26dd455ae59554b (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/44398"],"dc:language":["en"],"dc:rights":["Copyright 2013 Joseph Vandehey"],"dc:subject":["Asymptotic analysis","exponential sum","trigonometric sum","van der Corput transform"],"dc:title":["Error term improvements for van der Corput transforms"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:34Z"}