{"id":"https://openalex.org/W3134194363","doi":"https://doi.org/10.1109/mcse.2021.3065027","title":"Relationships Between the Zeros, Weights, and Weight Functions of Orthogonal Polynomials: Derivative Rule Conjecture (the DRC) Applied to Stieltjes and Spectral Imaging","display_name":"Relationships Between the Zeros, Weights, and Weight Functions of Orthogonal Polynomials: Derivative Rule Conjecture (the DRC) Applied to Stieltjes and Spectral Imaging","publication_year":2021,"publication_date":"2021-03-09","ids":{"openalex":"https://openalex.org/W3134194363","doi":"https://doi.org/10.1109/mcse.2021.3065027","mag":"3134194363"},"language":"en","primary_location":{"id":"doi:10.1109/mcse.2021.3065027","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mcse.2021.3065027","pdf_url":null,"source":{"id":"https://openalex.org/S107923245","display_name":"Computing in Science & Engineering","issn_l":"1521-9615","issn":["1521-9615","1558-366X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320061","host_organization_name":"AIP Publishing","host_organization_lineage":["https://openalex.org/P4310320061","https://openalex.org/P4310320257"],"host_organization_lineage_names":["AIP Publishing","American Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computing in Science &amp; Engineering","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/A5088145641","display_name":"William P. Reinhardt","orcid":"https://orcid.org/0000-0003-1944-1466"},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"William Reinhardt","raw_affiliation_strings":["University of Washington, Seattle, WA, USA"],"affiliations":[{"raw_affiliation_string":"University of Washington, Seattle, WA, USA","institution_ids":["https://openalex.org/I201448701"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5088145641"],"corresponding_institution_ids":["https://openalex.org/I201448701"],"apc_list":null,"apc_paid":null,"fwci":0.34,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.71508928,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"23","issue":"3","first_page":"56","last_page":"64"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12019","display_name":"Calibration and Measurement Techniques","score":0.9819999933242798,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12019","display_name":"Calibration and Measurement Techniques","score":0.9819999933242798,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace 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/T10640","display_name":"Spectroscopy and Chemometric Analyses","score":0.9749000072479248,"subfield":{"id":"https://openalex.org/subfields/1602","display_name":"Analytical Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.973800003528595,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6912797689437866},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.5542452335357666},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5467423796653748},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5441535115242004},{"id":"https://openalex.org/keywords/gaussian-quadrature","display_name":"Gaussian quadrature","score":0.5067394375801086},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.47822636365890503},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4280085563659668},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.33208930492401123},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.29619282484054565},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.18135079741477966},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.08642011880874634},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.07323771715164185}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6912797689437866},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.5542452335357666},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5467423796653748},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5441535115242004},{"id":"https://openalex.org/C167196314","wikidata":"https://www.wikidata.org/wiki/Q767680","display_name":"Gaussian quadrature","level":4,"score":0.5067394375801086},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.47822636365890503},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4280085563659668},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.33208930492401123},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.29619282484054565},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.18135079741477966},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.08642011880874634},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.07323771715164185},{"id":"https://openalex.org/C48265008","wikidata":"https://www.wikidata.org/wiki/Q7071295","display_name":"Nystr\u00f6m method","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/mcse.2021.3065027","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mcse.2021.3065027","pdf_url":null,"source":{"id":"https://openalex.org/S107923245","display_name":"Computing in Science & Engineering","issn_l":"1521-9615","issn":["1521-9615","1558-366X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320061","host_organization_name":"AIP Publishing","host_organization_lineage":["https://openalex.org/P4310320061","https://openalex.org/P4310320257"],"host_organization_lineage_names":["AIP Publishing","American Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computing in Science &amp; Engineering","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3437807525","display_name":null,"funder_award_id":"SB1341-12-CQ-0011","funder_id":"https://openalex.org/F4320332178","funder_display_name":"National Institute of Standards and Technology"}],"funders":[{"id":"https://openalex.org/F4320332178","display_name":"National Institute of Standards and Technology","ror":"https://ror.org/05xpvk416"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1963578603","https://openalex.org/W1966268321","https://openalex.org/W1976256098","https://openalex.org/W1983762405","https://openalex.org/W2102591969","https://openalex.org/W2170201775","https://openalex.org/W2320322137","https://openalex.org/W2485360899","https://openalex.org/W2980671223","https://openalex.org/W4245718566","https://openalex.org/W4249091683","https://openalex.org/W6902038016"],"related_works":["https://openalex.org/W2030322846","https://openalex.org/W2089411539","https://openalex.org/W2091267724","https://openalex.org/W1973321202","https://openalex.org/W3159111101","https://openalex.org/W1987395535","https://openalex.org/W4210699452","https://openalex.org/W1986739369","https://openalex.org/W2056088350","https://openalex.org/W2929787118"],"abstract_inverted_index":{"Zeros,":[0],"x[":[1,59],"k,":[2],"N":[3,72],"],":[4],"k":[5],"=":[6,86],"1":[7],"\u22efN,x[k,N],":[8],"k=1...N,":[9],"of":[10,42,53,67,123,137],"orthogonal":[11],"polynomials":[12],"are":[13,34],"useful":[14],"for":[15,99],"numerical":[16],"applications,":[17],"e.g.,":[18],"Gaussian":[19],"quadrature":[20,24],"were":[21],"combined":[22],"with":[23,30,114],"weights,":[25],"w[":[26,46],"k,N":[27,47,49,60,84,88,105,107],"],w[k,N],":[28],"integrals":[29],"weight":[31],"function":[32],"\u03c1(x)\u03c1(x)":[33],"performed":[35],"to":[36,120,129],"high":[37],"precision.":[38],"For":[39],"broad":[40],"classes":[41],"OPs,":[43],"the":[44,54,121,135,143],"ratios":[45,78],"]/\u03c1(x[":[48],"])":[50],"w[k,N]/\u03c1(x[k,N])are":[51],"independent":[52],"OPs":[55],"at":[56],"hand":[57],"if":[58],"]":[61,85],"x[k,N]lie":[62],"within":[63],"a":[64],"smooth":[65],"part":[66],"\u03c1(x)\u03c1(x).":[68],"This":[69,90],"universality":[70],"as":[71,82],"\u2192":[73],"\u221e":[74],"N\u2192\u221esuggests":[75],"that":[76,134],"these":[77],"may":[79],"be":[80],"evaluated":[81],"x'[":[83],"d(x[":[87],"])/dkx'[k,N]=d(x[k,N])/dk.":[89],"Derivative":[91],"Rule":[92],"Conjecture,":[93],"or":[94,112],"DRC,":[95,144],"then":[96],"gives,":[97],"even":[98],"small":[100],"NN,":[101],"\u03c1(x[":[102],"K,N":[103],"])=w[":[104],"]/x'[":[106],"],\u03c1(x[K,N])=w[k,N]/x'[k,N],":[108],"and":[109,141],"Stieltjes":[110],"imaging,":[111],"inversion,":[113],"exponential":[115],"convergence.":[116],"Similar":[117],"methods":[118],"apply":[119],"numerics":[122],"Schr\u00f6dinger":[124],"resolvents.":[125],"The":[126],"example":[127],"chosen":[128],"illustrate":[130],"this":[131],"latter":[132],"indicates":[133],"assumption":[136],"universality,":[138],"while":[139],"suggesting":[140],"validating":[142],"is":[145],"not":[146],"necessary.":[147]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
