{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:268"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:268","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions","abstract":"It is a classical result that for a function \\(f\\) \\(\\in\\) L\\(^p\\)(\\(\\char{bbold10}{0x54}\\)), dyadic partial sums of the Fourier series of \\(f\\) converge almost everywhere for \\(p\\) \\(\\in\\) (1, \\(\\infty\\)). In 1968, E. A. Bredihina established an analogous result for the Stepanov spaces of almost periodic functions in the case \\(p\\) = 2. Here, a new proof of the almost everywhere convergence result for Stepanov spaces is presented by way of a bound on an appropriate maximal operator for \\(p\\) = 2\\(^k\\), \\(k\\) \\(\\in\\) \\(\\char{bbold10}{0x4E}\\). In the process of establishing this, a number of general results are obtained that will facilitate further work pertaining to operator bounds and convergence issues in Stepanov spaces.","abstract_html":"It is a classical result that for a function \\(f\\) \\(\\in\\) L<span class=\"etd-inline-math\"><sup>p</sup></span>(\\(\\char{bbold10}{0x54}\\)), dyadic partial sums of the Fourier series of \\(f\\) converge almost everywhere for \\(p\\) \\(\\in\\) (1, \\(\\infty\\)). In 1968, E. A. Bredihina established an analogous result for the Stepanov spaces of almost periodic functions in the case \\(p\\) = 2. Here, a new proof of the almost everywhere convergence result for Stepanov spaces is presented by way of a bound on an appropriate maximal operator for \\(p\\) = 2<span class=\"etd-inline-math\"><sup>k</sup></span>, \\(k\\) \\(\\in\\) \\(\\char{bbold10}{0x4E}\\). In the process of establishing this, a number of general results are obtained that will facilitate further work pertaining to operator bounds and convergence issues in Stepanov spaces.","abstract_has_math":true,"creators":["Bailey, Andrew David"],"institution":"University of Birmingham","degree_name":"m_ph","degree_level":"m_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-07","date_published":"2009-07","updated_at":"2026-07-24T01:10:50Z","subjects":["QA Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://etheses.bham.ac.uk//id/eprint/268/2/Decl_IS_Bailey09MPhil.jpg","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["na"]},{"key":"dc:creator","label":"Author","values":["Bailey, Andrew David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2009-07"]},{"key":"dc:date.issued","label":"Date","values":["2009-07"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["College of Engineering & Physical Sciences","School of Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Birmingham"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["http://etheses.bham.ac.uk//id/eprint/268/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["m_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["m_ph"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://etheses.bham.ac.uk//id/eprint/268/1/Bailey09MPhil.pdf","http://etheses.bham.ac.uk//id/eprint/268/2/Decl_IS_Bailey09MPhil.jpg","http://etheses.bham.ac.uk//id/eprint/268/3/Bailey09MPhil_addendum.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It is a classical result that for a function \\(f\\) \\(\\in\\) L\\(^p\\)(\\(\\char{bbold10}{0x54}\\)), dyadic partial sums of the Fourier series of \\(f\\) converge almost everywhere for \\(p\\) \\(\\in\\) (1, \\(\\infty\\)). In 1968, E. A. Bredihina established an analogous result for the Stepanov spaces of almost periodic functions in the case \\(p\\) = 2. Here, a new proof of the almost everywhere convergence result for Stepanov spaces is presented by way of a bound on an appropriate maximal operator for \\(p\\) = 2\\(^k\\), \\(k\\) \\(\\in\\) \\(\\char{bbold10}{0x4E}\\). In the process of establishing this, a number of general results are obtained that will facilitate further work pertaining to operator bounds and convergence issues in Stepanov spaces."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","image/jpeg"]},{"key":"dc:title","label":"Title","values":["Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions"]}]}],"canonical_facts":{"dc:contributor.sponsor":["na"],"dc:creator":["Bailey, Andrew David"],"dc:date":["2009-07"],"dc:date.issued":["2009-07"],"dc:description.abstract":["It is a classical result that for a function \\(f\\) \\(\\in\\) L\\(^p\\)(\\(\\char{bbold10}{0x54}\\)), dyadic partial sums of the Fourier series of \\(f\\) converge almost everywhere for \\(p\\) \\(\\in\\) (1, \\(\\infty\\)). In 1968, E. A. Bredihina established an analogous result for the Stepanov spaces of almost periodic functions in the case \\(p\\) = 2. Here, a new proof of the almost everywhere convergence result for Stepanov spaces is presented by way of a bound on an appropriate maximal operator for \\(p\\) = 2\\(^k\\), \\(k\\) \\(\\in\\) \\(\\char{bbold10}{0x4E}\\). In the process of establishing this, a number of general results are obtained that will facilitate further work pertaining to operator bounds and convergence issues in Stepanov spaces."],"dc:format":["application/pdf","image/jpeg"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/268/1/Bailey09MPhil.pdf","http://etheses.bham.ac.uk//id/eprint/268/2/Decl_IS_Bailey09MPhil.jpg","http://etheses.bham.ac.uk//id/eprint/268/3/Bailey09MPhil_addendum.pdf"],"dc:publisher.department":["College of Engineering & Physical Sciences","School of Mathematics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/268/"],"dc:subject":["QA Mathematics"],"dc:title":["Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["m_ph"],"dc:type.qualificationname":["m_ph"]},"updated_at":"2026-07-24T01:10:50Z"}