{"id":"https://openalex.org/W4293498902","doi":"https://doi.org/10.1109/msp.2022.3172620","title":"Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes]","display_name":"Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes]","publication_year":2022,"publication_date":"2022-08-29","ids":{"openalex":"https://openalex.org/W4293498902","doi":"https://doi.org/10.1109/msp.2022.3172620"},"language":"en","primary_location":{"id":"doi:10.1109/msp.2022.3172620","is_oa":false,"landing_page_url":"https://doi.org/10.1109/msp.2022.3172620","pdf_url":null,"source":{"id":"https://openalex.org/S120977877","display_name":"IEEE Signal Processing Magazine","issn_l":"1053-5888","issn":["1053-5888","1558-0792"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Magazine","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100770168","display_name":"Pushpendra Singh","orcid":"https://orcid.org/0000-0001-5615-519X"},"institutions":[{"id":"https://openalex.org/I36909309","display_name":"National Institute of Technology Hamirpur","ror":"https://ror.org/01nc8zs04","country_code":"BD","type":"education","lineage":["https://openalex.org/I36909309","https://openalex.org/I4210152752"]}],"countries":["BD"],"is_corresponding":false,"raw_author_name":"Pushpendra Singh","raw_affiliation_strings":["Department of Electronics and Communication Engineering, National Institute of Technology Hamirpur, Hamirpur, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronics and Communication Engineering, National Institute of Technology Hamirpur, Hamirpur, India","institution_ids":["https://openalex.org/I36909309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086991902","display_name":"Amit Singhal","orcid":"https://orcid.org/0000-0002-4010-6614"},"institutions":[{"id":"https://openalex.org/I36090812","display_name":"Netaji Subhas University of Technology","ror":"https://ror.org/01fczmh85","country_code":"IN","type":"education","lineage":["https://openalex.org/I36090812"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Amit Singhal","raw_affiliation_strings":["Department of Electronics and Communication, Netaji Subhas University of Technology, Delhi, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronics and Communication, Netaji Subhas University of Technology, Delhi, India","institution_ids":["https://openalex.org/I36090812"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046768133","display_name":"Binish Fatimah","orcid":"https://orcid.org/0000-0002-6115-1561"},"institutions":[{"id":"https://openalex.org/I4210104573","display_name":"CMR University","ror":"https://ror.org/015waqy33","country_code":"IN","type":"education","lineage":["https://openalex.org/I4210104573"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Binish Fatimah","raw_affiliation_strings":["Department of Electronics and Communication Engineering, CMR Institute of Technology, Bangalore, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronics and Communication Engineering, CMR Institute of Technology, Bangalore, India","institution_ids":["https://openalex.org/I4210104573"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057412604","display_name":"Anubha Gupta","orcid":"https://orcid.org/0000-0002-7752-1926"},"institutions":[{"id":"https://openalex.org/I119939252","display_name":"Indraprastha Institute of Information Technology Delhi","ror":"https://ror.org/03vfp4g33","country_code":"IN","type":"education","lineage":["https://openalex.org/I119939252"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Anubha Gupta","raw_affiliation_strings":["Department of Electronics and Communication Engineering, Indraprastha Institute of Information Technology, Delhi, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronics and Communication Engineering, Indraprastha Institute of Information Technology, Delhi, India","institution_ids":["https://openalex.org/I119939252"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5107655888","display_name":"Shiv Dutt Joshi","orcid":"https://orcid.org/0000-0002-0326-0524"},"institutions":[{"id":"https://openalex.org/I68891433","display_name":"Indian Institute of Technology Delhi","ror":"https://ror.org/049tgcd06","country_code":"IN","type":"education","lineage":["https://openalex.org/I68891433"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Shiv Dutt Joshi","raw_affiliation_strings":["Department of Electrical Engineering, Indian Institute of Technology, Delhi, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Indian Institute of Technology, Delhi, India","institution_ids":["https://openalex.org/I68891433"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4144,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.56725942,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"39","issue":"5","first_page":"77","last_page":"84"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9768000245094299,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9684000015258789,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/maxima-and-minima","display_name":"Maxima and minima","score":0.6604006290435791},{"id":"https://openalex.org/keywords/fourier-series","display_name":"Fourier series","score":0.6398512125015259},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.6009089946746826},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5291006565093994},{"id":"https://openalex.org/keywords/classification-of-discontinuities","display_name":"Classification of discontinuities","score":0.5010321140289307},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.49433350563049316},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4543262720108032},{"id":"https://openalex.org/keywords/fourier-analysis","display_name":"Fourier analysis","score":0.440408855676651},{"id":"https://openalex.org/keywords/maxima","display_name":"Maxima","score":0.41519683599472046},{"id":"https://openalex.org/keywords/fourier-transform-on-finite-groups","display_name":"Fourier transform on finite groups","score":0.41178107261657715},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4112274646759033},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3841095566749573},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38183730840682983},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.33394429087638855},{"id":"https://openalex.org/keywords/short-time-fourier-transform","display_name":"Short-time Fourier transform","score":0.12702205777168274}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.6604006290435791},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.6398512125015259},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.6009089946746826},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5291006565093994},{"id":"https://openalex.org/C15627037","wikidata":"https://www.wikidata.org/wiki/Q541961","display_name":"Classification of discontinuities","level":2,"score":0.5010321140289307},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.49433350563049316},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4543262720108032},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.440408855676651},{"id":"https://openalex.org/C91528185","wikidata":"https://www.wikidata.org/wiki/Q520952","display_name":"Maxima","level":3,"score":0.41519683599472046},{"id":"https://openalex.org/C50826107","wikidata":"https://www.wikidata.org/wiki/Q5475803","display_name":"Fourier transform on finite groups","level":5,"score":0.41178107261657715},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4112274646759033},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3841095566749573},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38183730840682983},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33394429087638855},{"id":"https://openalex.org/C166386157","wikidata":"https://www.wikidata.org/wiki/Q1477735","display_name":"Short-time Fourier transform","level":4,"score":0.12702205777168274},{"id":"https://openalex.org/C554144382","wikidata":"https://www.wikidata.org/wiki/Q213156","display_name":"Performance art","level":2,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C52119013","wikidata":"https://www.wikidata.org/wiki/Q50637","display_name":"Art history","level":1,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/msp.2022.3172620","is_oa":false,"landing_page_url":"https://doi.org/10.1109/msp.2022.3172620","pdf_url":null,"source":{"id":"https://openalex.org/S120977877","display_name":"IEEE Signal Processing Magazine","issn_l":"1053-5888","issn":["1053-5888","1558-0792"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Magazine","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W947759089","https://openalex.org/W1430048851","https://openalex.org/W1506809288","https://openalex.org/W1574567355","https://openalex.org/W2013076189","https://openalex.org/W2047301883","https://openalex.org/W2054777142","https://openalex.org/W3102382472","https://openalex.org/W3125221504","https://openalex.org/W3159519317","https://openalex.org/W6809125596"],"related_works":["https://openalex.org/W2165559438","https://openalex.org/W2059850051","https://openalex.org/W4323309137","https://openalex.org/W4312337410","https://openalex.org/W2020016474","https://openalex.org/W4300635433","https://openalex.org/W1730415915","https://openalex.org/W4312855136","https://openalex.org/W2750615968","https://openalex.org/W4293498902"],"abstract_inverted_index":{"Fourier":[0,32,36,40],"theory":[1],"is":[2,193,221],"the":[3,92,127,133,160,165,183,188,197,201,209],"backbone":[4],"of":[5,21,56,94,106,117,119,126,135,139,151,185,204,211],"signal":[6,81],"processing":[7],"(SP)":[8],"and":[9,23,39,108,141,147,213],"communication":[10],"engineering.":[11],"It":[12],"has":[13],"been":[14],"widely":[15],"used":[16],"in":[17,25,65,71,182,218],"almost":[18],"all":[19],"branches":[20],"science":[22],"engineering":[24],"numerous":[26],"applications":[27],"since":[28],"its":[29],"inception.":[30],"However,":[31,122],"representations":[33],"such":[34],"as":[35,69,171,216],"series":[37],"(FS)":[38],"transform":[41],"(FT)":[42],"may":[43],"not":[44,172],"exist":[45],"for":[46,91,187],"some":[47,162],"signals":[48,163],"that":[49],"fail":[50],"to":[51,82,159,195,223],"fulfill":[52],"a":[53,62,80,102,113,123,136,148],"predefined":[54],"set":[55],"Dirichlet":[57],"conditions":[58,68],"(DCs).":[59],"We":[60],"note":[61],"subtle":[63],"gap":[64],"explaining":[66],"these":[67],"available":[70],"popular":[72],"SP":[73,219],"literature.":[74],"For":[75],"example,":[76],"original":[77,166],"DCs":[78,167,186],"require":[79],"have":[83],"bounded":[84],"variations":[85],"(BVs)":[86],"over":[87,143,154],"one":[88,144,155],"time":[89,145,156],"period":[90],"convergence":[93],"FS,":[95],"where":[96],"there":[97],"can":[98],"be,":[99],"at":[100,111],"most,":[101,112],"countably":[103,114],"infinite":[104,115],"number":[105,116,138,150],"maxima":[107,140],"minima":[109,142],"and,":[110],"discontinuities":[118,153],"finite":[120,137,149,152,202],"magnitude.":[121],"large":[124],"body":[125],"literature":[128],"replaces":[129],"this":[130],"statement":[131],"with":[132,200],"requirements":[134],"period,":[146],"period.":[157],"Due":[158],"latter,":[161],"fulfilling":[164],"are":[168],"incorrectly":[169],"perceived":[170],"having":[173],"convergent":[174],"FS":[175,205],"(CFS)":[176],"representation.":[177],"A":[178],"similar":[179],"problem":[180],"holds":[181],"description":[184],"FT.":[189],"Likewise,":[190],"although":[191],"it":[192],"easy":[194],"relate":[196],"first":[198],"DC":[199],"value":[203],"or":[206],"FT":[207],"coefficients,":[208],"relevance":[210],"second":[212],"third":[214],"DCs,":[215],"specified":[217],"literature,":[220],"hard":[222],"assimilate.":[224]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-08-29T07:29:34.045763","created_date":"2025-10-10T00:00:00"}