Caracterización de marcos asociados a un operador acotado en espacios de Hilbert
Se estudian algunos tipos de marcos asociados a ciertos operadores y ver las principales propiedades que se preservan de un marco, además de caracterizar los operadores de síntesis y marcos entre otros. Más aún el objetivo principal de estudiar marcos es reconstruir cada elemento de un espacio de Hi...
- Autores:
-
Monterrosa Castillo, Gleimer Enrique
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de Córdoba
- Repositorio:
- Repositorio Institucional Unicórdoba
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unicordoba.edu.co:ucordoba/8861
- Acceso en línea:
- https://repositorio.unicordoba.edu.co/handle/ucordoba/8861
https://repositorio.unicordoba.edu.co/
- Palabra clave:
- Base ortonormal
Espacio de Hilbert
Marcos
Sistemas atómicos
Orthonormal basis
Hilbert space
Frames
Atomic systems
- Rights
- openAccess
- License
- Copyright Universidad de Córdoba, 2025
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Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| title |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| spellingShingle |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert Base ortonormal Espacio de Hilbert Marcos Sistemas atómicos Orthonormal basis Hilbert space Frames Atomic systems |
| title_short |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| title_full |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| title_fullStr |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| title_full_unstemmed |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| title_sort |
Caracterización de marcos asociados a un operador acotado en espacios de Hilbert |
| dc.creator.fl_str_mv |
Monterrosa Castillo, Gleimer Enrique |
| dc.contributor.advisor.none.fl_str_mv |
Pastrana, Juan Carlos Villar Ferrer, Osmin |
| dc.contributor.author.none.fl_str_mv |
Monterrosa Castillo, Gleimer Enrique |
| dc.contributor.subjectmatterexpert.none.fl_str_mv |
Ferrer Villar, Osmin |
| dc.contributor.jury.none.fl_str_mv |
Pérez Reyes, Edgardo Lloreda Zuñiga, Jimmy |
| dc.subject.proposal.spa.fl_str_mv |
Base ortonormal Espacio de Hilbert Marcos Sistemas atómicos |
| topic |
Base ortonormal Espacio de Hilbert Marcos Sistemas atómicos Orthonormal basis Hilbert space Frames Atomic systems |
| dc.subject.keywords.eng.fl_str_mv |
Orthonormal basis Hilbert space Frames Atomic systems |
| description |
Se estudian algunos tipos de marcos asociados a ciertos operadores y ver las principales propiedades que se preservan de un marco, además de caracterizar los operadores de síntesis y marcos entre otros. Más aún el objetivo principal de estudiar marcos es reconstruir cada elemento de un espacio de Hilbert mediante una secuencia de imágenes de un operador lineal acotado. |
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2024 |
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2024-12-20 |
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2025-01-16T16:51:38Z |
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2025-01-16T16:51:38Z |
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Trabajo de grado - Pregrado |
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info:eu-repo/semantics/bachelorThesis |
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http://purl.org/coar/resource_type/c_7a1f |
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Universidad de Córdoba |
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Repositorio Universidad de Córdoba |
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https://repositorio.unicordoba.edu.co/ |
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spa |
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Aliprantis C, Burkinshaw O. Principles of real analysis. Gulf. Prof. Publis. (1998). Bartle, R., Sherbert, D. Introduction to real analysis. New York: Wiley, 2000. Casazza P, Deguang H, Larson D. Frames for Banach spaces, Contemporary Math. (1999) 149-182. Christensen O. An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, (2003). Duffin R, Schaeffer A. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, (1952), 341-366. Douglas R. On majorization, factorization and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17, (1966), 413-415. Daubechies I, Grossmann A, Meyer Y. Painless nonorthogonal expansions. J. Math. Phys. 27, (1986), 1271-1283. Esmeral K, Ferrer O, Wagner E, Frames in Krein spaces arising from a non regular W-metric, Banach J. Math. Anal. 9, (2015), 1-16. Ferrer O, Domínguez J, Arroyo E. Frames associated with an operator in spaces with an indefinite metric. AIMS Math. (2023), vol. 8, no 7, pág. 15712-15722. Ferrer O, Arroyo E, Naranjo J. Sistemas atómicos en espacios de Krein. Turkish J. Math. (2023), vol. 47, no 5, p. 1335-1349. Feichtinger H, Werther T. Atomic systems for subspaces, Proc. SampTA, (2001): 163-165. Găvruţa L. Frames for operator, Appl. Comput. Harmon. Anal. 32, (2012), 139- 144. Gröchenig K. Foundations of time-frequency analysis. Birkhäuser, Boston, (2001). Harro H. Functional Analysis Wiley, (1982). Kreyszig E. Introductory functional analysis with applications, J. Wiley & Sons. Inc. (1978). Kovacevic J, Chebira A. An introduction to frames, Found. Tren. Sign. Proc. 2 (2008) (1) 1-94. Mohammed A, Samir K, Bounader N. K-frames for Krein spaces. Ann. Funct. Anal. 14, (2023), 1-20. Marina H, Matan G, Dustin G, Ram Z. Asymptotic Frame Theory for Analog Coding. Found. Tren. Comm. Inf. Theo. (2021) vol. 18, no. 4, pp. 526-645. Paulo S, Diniz, Johan A, Suykens, Rama C. y Sergios T. Frames in Signal Pro cessing. Aca. Press. Lib. Sig. Proc. (2014) Vol. 1 Pg. 561-5 Rajupillai k, Palaniammals. Frame Theory and Application in Digital Image Pro cessing. Int. J. Lat. Tren. Engin. Tech. (2015.) Vol. 6 6. Rudin, Walter. Análisis funcional. Reverté, 2012. Sitati I, Musundi S, Nzimbi B, Dennis K. A Note On Quasi-Similarity of Ope rators in Hilbert Spaces. Res. Conf. hel. Main. Cam. from 28 th–30th October, (2015) p. 356. Vera A, Alegría P. Un curso de análisis funcional. (1997). Xiao X, Zhu Y, Găvruţa L. Some properties of K-frames in Hilbert spaces. Results Math. 63, (2013) no 3-4, p. 1243-1255. Yosida K, Functional analysis, Sprin. Verl. (1965). |
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Copyright Universidad de Córdoba, 2025 |
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Montería, Córdoba, Colombia |
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Pastrana, Juan Carlosd5d1b39b-85fa-4354-b126-0cf5ea03a808600Villar Ferrer, Osmin016399ae-f814-4731-9d9f-ca4a549e0803-1Monterrosa Castillo, Gleimer Enrique912f15ba-3b7c-40f7-ae5d-cf55acd8dd14-1Ferrer Villar, OsminPérez Reyes, Edgardo2f7a4087-2199-45d8-a178-85305dcc245f-1Lloreda Zuñiga, Jimmy60af3cdc-b9cd-4332-84b0-60abbcc08fe2-12025-01-16T16:51:38Z2025-01-16T16:51:38Z2024-12-20https://repositorio.unicordoba.edu.co/handle/ucordoba/8861Universidad de CórdobaRepositorio Universidad de Córdobahttps://repositorio.unicordoba.edu.co/Se estudian algunos tipos de marcos asociados a ciertos operadores y ver las principales propiedades que se preservan de un marco, además de caracterizar los operadores de síntesis y marcos entre otros. Más aún el objetivo principal de estudiar marcos es reconstruir cada elemento de un espacio de Hilbert mediante una secuencia de imágenes de un operador lineal acotado.The study focuses on certain types of frames associated with specific operators and examines the main properties that are preserved in a frame. Additionally, it aims to characterize synthesis operators and frames, among others. Furthermore, the main objective of studying frames is to reconstruct each element of a Hilbert space through a sequence of images of a bounded linear operator.Fundamentos de Análisis Funcional (página 2)Operadores lineales en espacios de Hilbert (página 5)Marcos en espacios de Hilbert (página 12)Sucesiones de Bessel en espacios de Hilbert (página 12)Marcos en espacios de Hilbert (página 14)Marcos duales (página 24)Marcos asociados a un operador en espacios de Hilbert (página 28)Sistemas atómicos (página 28)Algunos tipos de marcos asociados a un operador (página 28)Bibliografía (página 28)PregradoMatemático(a)Monografíasapplication/pdfspaUniversidad de CórdobaFacultad de Ciencias BásicasMontería, Córdoba, ColombiaMatemáticaCopyright Universidad de Córdoba, 2025https://creativecommons.org/licenses/by-nc-nd/4.0/Atribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Caracterización de marcos asociados a un operador acotado en espacios de HilbertTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesishttp://purl.org/coar/resource_type/c_7a1finfo:eu-repo/semantics/acceptedVersionTextAliprantis C, Burkinshaw O. Principles of real analysis. Gulf. Prof. Publis. (1998).Bartle, R., Sherbert, D. Introduction to real analysis. New York: Wiley, 2000.Casazza P, Deguang H, Larson D. Frames for Banach spaces, Contemporary Math. (1999) 149-182.Christensen O. An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, (2003).Duffin R, Schaeffer A. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, (1952), 341-366.Douglas R. On majorization, factorization and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17, (1966), 413-415.Daubechies I, Grossmann A, Meyer Y. Painless nonorthogonal expansions. J. Math. Phys. 27, (1986), 1271-1283.Esmeral K, Ferrer O, Wagner E, Frames in Krein spaces arising from a non regular W-metric, Banach J. Math. Anal. 9, (2015), 1-16.Ferrer O, Domínguez J, Arroyo E. Frames associated with an operator in spaces with an indefinite metric. AIMS Math. (2023), vol. 8, no 7, pág. 15712-15722.Ferrer O, Arroyo E, Naranjo J. Sistemas atómicos en espacios de Krein. Turkish J. Math. (2023), vol. 47, no 5, p. 1335-1349.Feichtinger H, Werther T. Atomic systems for subspaces, Proc. SampTA, (2001): 163-165.Găvruţa L. Frames for operator, Appl. Comput. Harmon. Anal. 32, (2012), 139- 144.Gröchenig K. Foundations of time-frequency analysis. Birkhäuser, Boston, (2001).Harro H. Functional Analysis Wiley, (1982).Kreyszig E. Introductory functional analysis with applications, J. Wiley & Sons. Inc. (1978).Kovacevic J, Chebira A. An introduction to frames, Found. Tren. Sign. Proc. 2 (2008) (1) 1-94.Mohammed A, Samir K, Bounader N. K-frames for Krein spaces. Ann. Funct. Anal. 14, (2023), 1-20.Marina H, Matan G, Dustin G, Ram Z. Asymptotic Frame Theory for Analog Coding. Found. Tren. Comm. Inf. Theo. (2021) vol. 18, no. 4, pp. 526-645.Paulo S, Diniz, Johan A, Suykens, Rama C. y Sergios T. Frames in Signal Pro cessing. Aca. Press. Lib. Sig. Proc. (2014) Vol. 1 Pg. 561-5Rajupillai k, Palaniammals. Frame Theory and Application in Digital Image Pro cessing. Int. J. Lat. Tren. Engin. Tech. (2015.) Vol. 6 6.Rudin, Walter. Análisis funcional. Reverté, 2012.Sitati I, Musundi S, Nzimbi B, Dennis K. A Note On Quasi-Similarity of Ope rators in Hilbert Spaces. Res. Conf. hel. Main. Cam. from 28 th–30th October, (2015) p. 356.Vera A, Alegría P. Un curso de análisis funcional. (1997).Xiao X, Zhu Y, Găvruţa L. Some properties of K-frames in Hilbert spaces. Results Math. 63, (2013) no 3-4, p. 1243-1255.Yosida K, Functional analysis, Sprin. Verl. 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