When the Identity Theorem "
The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting if we relax the analyticity hypothesis on...
| Autores: | , , , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/50443 |
| Acceso en línea: | https://riunet.upv.es/handle/10251/50443 |
| Access Level: | acceso abierto |
| Palabra clave: | Lineability Spaceability Algebrability Analytic function Identity theorem Annulling function MATEMATICA APLICADA |
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When the Identity Theorem "seems"to failConejero, J. Alberto|||0000-0003-3681-7533Jiménez Rodríguez, PabloMuñoz-Fernández, Gustavo A.Seoane-Sepúlveda, Juan B.LineabilitySpaceabilityAlgebrabilityAnalytic functionIdentity theoremAnnulling functionMATEMATICA APLICADAThe Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting if we relax the analyticity hypothesis on the function to infinitely many times differentiability. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is c), (ii) every nonzero element of A is nowhere analytic, (iii) A ⊂ C∞(R), (iv) every element of A has infinitely many zeroes in R, and (v) for every f ∈ A and n ∈ N, f (n) (the n-th derivative of f) enjoys the same properties as the elements in A\{0}. This construction complements those made by Cater and Kim & Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectivelyThe authors would like to thank the referees for their insightful comments and remarks, especially for pointing out references [4, 14] and for their help in improving Theorem 2.3. They would also like to thank Profs. Bastin, Esser, and Nikolay for sharing their latest preprint [4]. The first, second, and fourth authors were supported by grants MTM2009-07848 and MTM2010-14909. The second named author's research was performed during the completion of his Master's Thesis at the Universidad Complutense de Madrid (Spain) in 2011/2012.Mathematical Association of America (MAA)Departamento de Matemática AplicadaInstituto Universitario de Matemática Pura y AplicadaEscuela Técnica Superior de Ingeniería InformáticaMinisterio de Ciencia e InnovaciónRepositorio Institucional de la Universitat Politècnica de València Riunet20142014-01-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfapplication/pdfhttps://riunet.upv.es/handle/10251/50443reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengMinisterio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 MTM2009-07848 Analisis Funcional No-Lineal Y GeometricoMinisterio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 MTM2010-14909 HIPERCICLICIDAD Y CAOS DE OPERADORESopen accesshttp://purl.org/coar/access_right/c_abf2Reserva de todos los derechoshttp://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/504432026-06-13T07:49:27Z |
| dc.title.none.fl_str_mv |
When the Identity Theorem " seems" to fail |
| title |
When the Identity Theorem " |
| spellingShingle |
When the Identity Theorem " Conejero, J. Alberto|||0000-0003-3681-7533 Lineability Spaceability Algebrability Analytic function Identity theorem Annulling function MATEMATICA APLICADA |
| title_short |
When the Identity Theorem " |
| title_full |
When the Identity Theorem " |
| title_fullStr |
When the Identity Theorem " |
| title_full_unstemmed |
When the Identity Theorem " |
| title_sort |
When the Identity Theorem " |
| dc.creator.none.fl_str_mv |
Conejero, J. Alberto|||0000-0003-3681-7533 Jiménez Rodríguez, Pablo Muñoz-Fernández, Gustavo A. Seoane-Sepúlveda, Juan B. |
| author |
Conejero, J. Alberto|||0000-0003-3681-7533 |
| author_facet |
Conejero, J. Alberto|||0000-0003-3681-7533 Jiménez Rodríguez, Pablo Muñoz-Fernández, Gustavo A. Seoane-Sepúlveda, Juan B. |
| author_role |
author |
| author2 |
Jiménez Rodríguez, Pablo Muñoz-Fernández, Gustavo A. Seoane-Sepúlveda, Juan B. |
| author2_role |
author author author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemática Aplicada Instituto Universitario de Matemática Pura y Aplicada Escuela Técnica Superior de Ingeniería Informática Ministerio de Ciencia e Innovación Repositorio Institucional de la Universitat Politècnica de València Riunet |
| dc.subject.none.fl_str_mv |
Lineability Spaceability Algebrability Analytic function Identity theorem Annulling function MATEMATICA APLICADA |
| topic |
Lineability Spaceability Algebrability Analytic function Identity theorem Annulling function MATEMATICA APLICADA |
| description |
The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting if we relax the analyticity hypothesis on the function to infinitely many times differentiability. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is c), (ii) every nonzero element of A is nowhere analytic, (iii) A ⊂ C∞(R), (iv) every element of A has infinitely many zeroes in R, and (v) for every f ∈ A and n ∈ N, f (n) (the n-th derivative of f) enjoys the same properties as the elements in A\{0}. This construction complements those made by Cater and Kim & Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively |
| publishDate |
2014 |
| dc.date.none.fl_str_mv |
2014 2014-01-01 |
| dc.type.none.fl_str_mv |
journal article http://purl.org/coar/resource_type/c_6501 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
https://riunet.upv.es/handle/10251/50443 |
| url |
https://riunet.upv.es/handle/10251/50443 |
| dc.language.none.fl_str_mv |
Inglés eng |
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Inglés |
| language |
eng |
| dc.relation.none.fl_str_mv |
Ministerio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 MTM2009-07848 Analisis Funcional No-Lineal Y Geometrico Ministerio de Ciencia e Innovación http://dx.doi.org/10.13039/501100004837 MTM2010-14909 HIPERCICLICIDAD Y CAOS DE OPERADORES |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 Reserva de todos los derechos http://rightsstatements.org/vocab/InC/1.0/ |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 Reserva de todos los derechos http://rightsstatements.org/vocab/InC/1.0/ |
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openAccess |
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application/pdf application/pdf |
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Mathematical Association of America (MAA) |
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Mathematical Association of America (MAA) |
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reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia instname:Universitat Politècnica de València (UPV) |
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Universitat Politècnica de València (UPV) |
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