{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86781"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86781","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Convergence in Ergodic Theory","abstract":"We investigate two problems involving convergence in ergodic theory. The first problem is the following: Given a measure preserving transformation T and a weight function w(alpha) → 0 as alpha → 0, is there a p > 0 such that the expression w(alpha)#{ n : 1n k=1nfT kx >a } have a limit, a.s. or in norm, as alpha → 0 for all functions f ∈ Lp0 [0,1]? No, we show. Here # denotes counting measure and f's are taken to be mean-zero functions. We also consider similar questions for the more general operator w(alpha)#{n : 1nq k=1n f(Tk(x)) > alpha}, q > 1. The second problem addressed is to give arithmetic and probabilistic characterizations on the integer sequence ( nk) such that the series of ergodic differences k=1infinity ( Ank+1f-Ankf ), where An denotes the usual ergodic averages, converges unconditionally for all functions f in some Lp space.","abstract_html":"We investigate two problems involving convergence in ergodic theory. The first problem is the following: Given a measure preserving transformation T and a weight function w(alpha) → 0 as alpha → 0, is there a p > 0 such that the expression w(alpha)#{ n : 1n k=1nfT kx >a } have a limit, a.s. or in norm, as alpha → 0 for all functions f ∈ Lp0 [0,1]? No, we show. Here # denotes counting measure and f's are taken to be mean-zero functions. We also consider similar questions for the more general operator w(alpha)#{n : 1nq k=1n f(Tk(x)) > alpha}, q > 1. The second problem addressed is to give arithmetic and probabilistic characterizations on the integer sequence ( nk) such that the series of ergodic differences k=1infinity ( Ank+1f-Ankf ), where An denotes the usual ergodic averages, converges unconditionally for all functions f in some Lp space.","abstract_has_math":false,"creators":["Argiris, Georgios"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rosenblatt, Joseph"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:30Z","date_published":"2015-09-28T15:19:30Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3023009"],"render_values":[{"text":"(MiAaPQ)AAI3023009","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86781","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rosenblatt, Joseph"]},{"key":"dc:creator","label":"Author","values":["Argiris, Georgios"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:30Z","10000-01-01","2001"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86781","(MiAaPQ)AAI3023009"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We investigate two problems involving convergence in ergodic theory. The first problem is the following: Given a measure preserving transformation T and a weight function w(alpha) → 0 as alpha → 0, is there a p > 0 such that the expression w(alpha)#{ n : 1n k=1nfT kx >a } have a limit, a.s. or in norm, as alpha → 0 for all functions f ∈ Lp0 [0,1]? No, we show. Here # denotes counting measure and f's are taken to be mean-zero functions. We also consider similar questions for the more general operator w(alpha)#{n : 1nq k=1n f(Tk(x)) > alpha}, q > 1. The second problem addressed is to give arithmetic and probabilistic characterizations on the integer sequence ( nk) such that the series of ergodic differences k=1infinity ( Ank+1f-Ankf ), where An denotes the usual ergodic averages, converges unconditionally for all functions f in some Lp space.","Made available in DSpace on 2015-09-28T15:19:30Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3023009.pdf: 2049317 bytes, checksum: 01968016ed63aca4d8c1d7ce4c221aa3 (MD5) Previous issue date: 2001","Embargo set by: Seth Robbins for item 88062 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","51 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001."]},{"key":"dc:title","label":"Title","values":["Convergence in Ergodic Theory"]}]}],"canonical_facts":{"dc:contributor":["Rosenblatt, Joseph"],"dc:creator":["Argiris, Georgios"],"dc:date":["2015-09-28T15:19:30Z","10000-01-01","2001"],"dc:description":["We investigate two problems involving convergence in ergodic theory. The first problem is the following: Given a measure preserving transformation T and a weight function w(alpha) → 0 as alpha → 0, is there a p > 0 such that the expression w(alpha)#{ n : 1n k=1nfT kx >a } have a limit, a.s. or in norm, as alpha → 0 for all functions f ∈ Lp0 [0,1]? No, we show. Here # denotes counting measure and f's are taken to be mean-zero functions. We also consider similar questions for the more general operator w(alpha)#{n : 1nq k=1n f(Tk(x)) > alpha}, q > 1. The second problem addressed is to give arithmetic and probabilistic characterizations on the integer sequence ( nk) such that the series of ergodic differences k=1infinity ( Ank+1f-Ankf ), where An denotes the usual ergodic averages, converges unconditionally for all functions f in some Lp space.","Made available in DSpace on 2015-09-28T15:19:30Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3023009.pdf: 2049317 bytes, checksum: 01968016ed63aca4d8c1d7ce4c221aa3 (MD5) Previous issue date: 2001","Embargo set by: Seth Robbins for item 88062 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","51 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001."],"dc:identifier":["http://hdl.handle.net/2142/86781","(MiAaPQ)AAI3023009"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Convergence in Ergodic Theory"],"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":"2026-07-22T22:26:27Z"}