{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21400"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21400","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Efficient quotient representation of meromorphic functions","abstract":"Representations of meromorphic functions as quotients of analytic functions have been studied for years. Miles showed that any meromorphic function f can be written as f$\\sb1$/f$\\sb2$ where each f$\\sb{\\rm j}$ is entire and T(r,f$\\sb{\\rm j}$) $\\leq$ AT(Br,f). This result is trivial if the pole set Z is finite. For an infinite pole set Z of f, he established the existence of entire f$\\sb{\\rm j}$ such that T(r,f$\\sb{\\rm j}$) $\\leq$ A$\\sp\\prime$N(B$\\sp\\prime$r,Z).","abstract_html":"Representations of meromorphic functions as quotients of analytic functions have been studied for years. Miles showed that any meromorphic function f can be written as f$\\sb1$/f$\\sb2$ where each f$\\sb{\\rm j}$ is entire and T(r,f$\\sb{\\rm j}$) $\\leq$ AT(Br,f). This result is trivial if the pole set Z is finite. For an infinite pole set Z of f, he established the existence of entire f$\\sb{\\rm j}$ such that T(r,f$\\sb{\\rm j}$) $\\leq$ A$\\sp\\prime$N(B$\\sp\\prime$r,Z).","abstract_has_math":true,"creators":["Hopkins, Kevin Walter"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kaufman, Robert"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:07:35Z","date_published":"2011-05-07T13:07:35Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Hopkins, Kevin Walter"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924841","(UMI)AAI8924841"],"render_values":[{"text":"AAI8924841","href":null,"code":true},{"text":"(UMI)AAI8924841","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21400","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kaufman, Robert"]},{"key":"dc:creator","label":"Author","values":["Hopkins, Kevin Walter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:07:35Z","10000-01-01","1989"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1989 Hopkins, Kevin Walter"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924841","(UMI)AAI8924841","http://hdl.handle.net/2142/21400"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Representations of meromorphic functions as quotients of analytic functions have been studied for years. Miles showed that any meromorphic function f can be written as f$\\sb1$/f$\\sb2$ where each f$\\sb{\\rm j}$ is entire and T(r,f$\\sb{\\rm j}$) $\\leq$ AT(Br,f). This result is trivial if the pole set Z is finite. For an infinite pole set Z of f, he established the existence of entire f$\\sb{\\rm j}$ such that T(r,f$\\sb{\\rm j}$) $\\leq$ A$\\sp\\prime$N(B$\\sp\\prime$r,Z).","Miles's technique, called balancing, was to add elements to the pole set Z in such a way that he could apply a result of Rubel and Taylor. Our technique is to add elements Z$\\sp\\prime$ and Z$\\sp{\\prime\\prime}$ to the pole set Z in such a manner as to make the Fourier coefficients of $\\rm\\log\\ \\vert f\\sb2(re\\sp{i\\theta})\\vert$ small. We need to ensure that the number of zeros added in the balancing does not make N(r, Z $\\cup$ Z$\\sp\\prime$ $\\cup$ Z$\\sp{\\prime\\prime}$) $\\gg$ N(r,Z). We also need to ensure that the zeros added to make one coefficient small do not adversely interact with other coefficients.","Miles exhibited a function where T(r,f$\\sb{\\rm j}$) $\\leq$ AT(r,f) is not possible on some sequence of r's. In this thesis we examine the cases where we can set the constant B to equal one in Miles's result.","We are able to achieve A = 1 + o(1) and B = 1 on a sequence of r$\\sb{\\rm n}$'s. For a meromorphic function f of finite order we achieve$$\\rm A = O(\\rho\\ \\max\\ \\left\\{1,{n(r,Z)\\over N(r,Z)}\\right\\})$$with B = 1 on a set of r's of positive logarithmic density. For a meromorphic function f of infinite order we obtain a more complicated expression for A with B = 1 on a set of positive logarithmic density.","Made available in DSpace on 2011-05-07T13:07:35Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924841.pdf: 3444108 bytes, checksum: 2e1cd04e94059b14160e057d1b03484d (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:31Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:05-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Efficient quotient representation of meromorphic functions"]}]}],"canonical_facts":{"dc:contributor":["Kaufman, Robert"],"dc:creator":["Hopkins, Kevin Walter"],"dc:date":["2011-05-07T13:07:35Z","10000-01-01","1989"],"dc:description":["Representations of meromorphic functions as quotients of analytic functions have been studied for years. Miles showed that any meromorphic function f can be written as f$\\sb1$/f$\\sb2$ where each f$\\sb{\\rm j}$ is entire and T(r,f$\\sb{\\rm j}$) $\\leq$ AT(Br,f). This result is trivial if the pole set Z is finite. For an infinite pole set Z of f, he established the existence of entire f$\\sb{\\rm j}$ such that T(r,f$\\sb{\\rm j}$) $\\leq$ A$\\sp\\prime$N(B$\\sp\\prime$r,Z).","Miles's technique, called balancing, was to add elements to the pole set Z in such a way that he could apply a result of Rubel and Taylor. Our technique is to add elements Z$\\sp\\prime$ and Z$\\sp{\\prime\\prime}$ to the pole set Z in such a manner as to make the Fourier coefficients of $\\rm\\log\\ \\vert f\\sb2(re\\sp{i\\theta})\\vert$ small. We need to ensure that the number of zeros added in the balancing does not make N(r, Z $\\cup$ Z$\\sp\\prime$ $\\cup$ Z$\\sp{\\prime\\prime}$) $\\gg$ N(r,Z). We also need to ensure that the zeros added to make one coefficient small do not adversely interact with other coefficients.","Miles exhibited a function where T(r,f$\\sb{\\rm j}$) $\\leq$ AT(r,f) is not possible on some sequence of r's. In this thesis we examine the cases where we can set the constant B to equal one in Miles's result.","We are able to achieve A = 1 + o(1) and B = 1 on a sequence of r$\\sb{\\rm n}$'s. For a meromorphic function f of finite order we achieve$$\\rm A = O(\\rho\\ \\max\\ \\left\\{1,{n(r,Z)\\over N(r,Z)}\\right\\})$$with B = 1 on a set of r's of positive logarithmic density. For a meromorphic function f of infinite order we obtain a more complicated expression for A with B = 1 on a set of positive logarithmic density.","Made available in DSpace on 2011-05-07T13:07:35Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924841.pdf: 3444108 bytes, checksum: 2e1cd04e94059b14160e057d1b03484d (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:31Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:05-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI8924841","(UMI)AAI8924841","http://hdl.handle.net/2142/21400"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Hopkins, Kevin Walter"],"dc:subject":["Mathematics"],"dc:title":["Efficient quotient representation of meromorphic functions"],"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":"2026-07-22T22:25:17Z"}