Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables
Texto, ecuaciones, fórmulas
- Autores:
-
Salazar Morales, Omar
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2022
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/81829
- Palabra clave:
- 510 - Matemáticas::515 - Análisis
Polinomios ortogonales
Polinomios Sobolev
Polinomios en varias variables
Productos internos
Productos internos Sobolev
Ecuaciones diferenciales
Ecuaciones diferenciales parciales
Orthogonal polynomials
Sobolev polynomials
Polynomials in several variables
Inner products
Sobolev inner products
Differential equations
Partial differential equations
- Rights
- openAccess
- License
- Atribución-NoComercial-SinDerivadas 4.0 Internacional
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| dc.title.spa.fl_str_mv |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| dc.title.translated.eng.fl_str_mv |
Properties of some Sobolev orthogonal polynomial systems in several variables |
| title |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| spellingShingle |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables 510 - Matemáticas::515 - Análisis Polinomios ortogonales Polinomios Sobolev Polinomios en varias variables Productos internos Productos internos Sobolev Ecuaciones diferenciales Ecuaciones diferenciales parciales Orthogonal polynomials Sobolev polynomials Polynomials in several variables Inner products Sobolev inner products Differential equations Partial differential equations |
| title_short |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| title_full |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| title_fullStr |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| title_full_unstemmed |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| title_sort |
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables |
| dc.creator.fl_str_mv |
Salazar Morales, Omar |
| dc.contributor.advisor.none.fl_str_mv |
Dueñas Ruiz, Herbert Alonso |
| dc.contributor.author.none.fl_str_mv |
Salazar Morales, Omar |
| dc.contributor.researchgroup.spa.fl_str_mv |
Grupo de Investigación en Polinomios Ortogonales y Aplicaciones |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas::515 - Análisis |
| topic |
510 - Matemáticas::515 - Análisis Polinomios ortogonales Polinomios Sobolev Polinomios en varias variables Productos internos Productos internos Sobolev Ecuaciones diferenciales Ecuaciones diferenciales parciales Orthogonal polynomials Sobolev polynomials Polynomials in several variables Inner products Sobolev inner products Differential equations Partial differential equations |
| dc.subject.proposal.spa.fl_str_mv |
Polinomios ortogonales Polinomios Sobolev Polinomios en varias variables Productos internos Productos internos Sobolev Ecuaciones diferenciales Ecuaciones diferenciales parciales |
| dc.subject.proposal.eng.fl_str_mv |
Orthogonal polynomials Sobolev polynomials Polynomials in several variables Inner products Sobolev inner products Differential equations Partial differential equations |
| description |
Texto, ecuaciones, fórmulas |
| publishDate |
2022 |
| dc.date.accessioned.none.fl_str_mv |
2022-08-09T21:04:55Z |
| dc.date.available.none.fl_str_mv |
2022-08-09T21:04:55Z |
| dc.date.issued.none.fl_str_mv |
2022-02-01 |
| dc.type.spa.fl_str_mv |
Trabajo de grado - Doctorado |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
| dc.type.version.spa.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| dc.type.coar.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_db06 |
| dc.type.content.spa.fl_str_mv |
Text |
| dc.type.redcol.spa.fl_str_mv |
http://purl.org/redcol/resource_type/TD |
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http://purl.org/coar/resource_type/c_db06 |
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acceptedVersion |
| dc.identifier.uri.none.fl_str_mv |
https://repositorio.unal.edu.co/handle/unal/81829 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad Nacional de Colombia |
| dc.identifier.reponame.spa.fl_str_mv |
Repositorio Institucional Universidad Nacional de Colombia |
| dc.identifier.repourl.spa.fl_str_mv |
https://repositorio.unal.edu.co/ |
| url |
https://repositorio.unal.edu.co/handle/unal/81829 https://repositorio.unal.edu.co/ |
| identifier_str_mv |
Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
| dc.language.iso.spa.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.spa.fl_str_mv |
Abramowitz, Milton and Stegun, Irene Ann, eds. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Applied mathematics series - 55. Washington, D.C.: National Bureau of Standards, June 1964. Aktas, Rabia and Xu, Yuan. “Sobolev orthogonal polynomials on a simplex”. In: Int. Math. Res. Not. IMRN 2013.13 (2012), pp. 3087–3131. Alfaro, Manuel, Alvarez de Morales, Maria, and Rezola, M. Luisa. “Orthogonality of the Jacobi polynomials with negative integer parameters”. In: J. Comput. Appl. Math. 145.2 (2002), pp. 379–386. Alfaro, Manuel, Lopez, Guillermo, and Rezola, M. Luisa. “Some properties of zeros of Sobolev-type orthogonal polynomials”. In: J. Comput. Appl. Math. 69.1 (1996), pp. 171–179. Alfaro, Manuel, Marcellan, Francisco, Peña, Ana, and Rezola, M. Luisa. “On linearly related orthogonal polynomials and their functionals”. In: J. Math. Anal. Appl. 287.1 (2003), pp. 307–319. Alfaro, Manuel, Perez, Teresa E., Piñar, Miguel A., and Rezola, M. Luisa. “Sobolev orthogonal polynomials: the discrete-continuous case”. In: Methods Appl. Anal. 6.4 (1999), pp. 593–616. Althammer, Peter. “Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation”. In: J. Reine Angew. Math. 211 (Apr. 1962), pp. 192–204. Alvarez de Morales, Maria, Perez, Teresa E., and Piñar, Miguel A. “Sobolev orthogonality for the Gegenbauer polynomials {C_n^{−N+1/2}}n≥0”. In: J. Comput. Appl. Math. 100.1 (1998), pp. 111–120. Alvarez-Nodarse, Renato. Polinomios hipergeométricos clásicos y q-polinomios. Monografías del seminario matemático García de Galdeano. Prensas Universitarias de Zaragoza. Departamento de Matemáticas, Universidad de Zaragoza, 2003. Alvarez-Nodarse, Renato and Marcellan, Francisco. “A generalization of the class Laguerre polynomials: asymptotic properties and zeros”. In: Appl. Anal. 62.3–4 (1996), pp. 349–366. Alvarez-Nodarse, Renato and Moreno-Balcazar, Juan Jose. “Asymptotic properties of generalized Laguerre orthogonal polynomials”. In: Indag. Math. (N.S.) 15.2 (2004), pp. 151–165. Arvesu, Jorge, Alvarez-Nodarse, Renato, Marcellan, Francisco, and Pan, Ke-Lin. “Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros”. In: J. Comput. Appl. Math. 90.2 (1998), pp. 135–156. Atkinson, Kendall and Hansen, Olaf. “Solving the nonlinear Poisson equation on the unit disk”. In: J. Integral Equations Appl. 17.3 (2005), pp. 223–241. Bavinck, Herman and Meijer, Hendrik Gerrit. “Orthogonal polynomials with respect to a symmetric inner product involving derivatives”. In: Appl. Anal. 33.1–2 (1989), pp. 103–117. Bavinck, Herman and Meijer, Hendrik Gerrit. “On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations”. In: Indag. Math. (N.S.) 1.1 (1990), pp. 7–14. Berti, Andrea C., Bracciali, Cleonice F., and Sri Ranga, Alagacone. “Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials”. In: Numer. Algorithms 34 (2003), pp. 203–216. Berti, Andrea C. and Sri Ranga, Alagacone. “Companion orthogonal polynomials: some applications”. In: Appl. Numer. Math. 39.2 (2001), pp. 127– 149. Bracciali, Cleonice F., Delgado, Antonia M., Fernandez, Lidia, Perez, Teresa E., and Piñar, Miguel A. “New steps on Sobolev orthogonality in two variables”. In: J. Comput. Appl. Math. 235.4 (2010), pp. 916–926. Brenner, J. “Über eine Erweiterung des Orthogonaltätsbegriffes bei Polynomen”. In: Proceedings of the Conference on Constructive Theory of Functions. Ed. by Alexits, G. and Stechkin, S. B. Budapest: Akademiai Kiado, Aug. 1972, pp. 77–83. Burris, Stanley and Sankappanavar, Hanamantagouda P. A Course in Universal Algebra. Vol. 78. Graduate Texts in Mathematics. New York: Springer- Verlag, 1981. Chihara, Theodore S. An introduction to orthogonal polynomials. Gordon and Breach, Science Publishers, Inc., 1978. Cohen, Edgar A. “Zero distribution and behavior of orthogonal polynomials in the Sobolev space W^{1,2}[-1,1]”. In: SIAM J. Math. Anal. 6.1 (1975), pp. 105– 116. Dai, Feng and Xu, Yuan. “Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball”. In: J. Approx. Theory 163.10 (2011), pp. 1400– 1418. Dai, Feng and Xu, Yuan. Approximation theory and harmonic analysis on spheres and balls. Springer monographs in mathematics. New York: Springer, 2013. De Bruin, Marcel G. “A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces”. In: J. Comput. Appl. Math. 49 (1993), pp. 27– 35. De Bruin, Marcel G. and Meijer, Hendrik Gerrit. “Zeros of orthogonal polynomials in a non-discrete Sobolev space”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 233–246. de Jesus, Marcio Dinis do Nascimento, Marcellan, Francisco, Petronilho, Jose Carlos, and Pinzon-Cortes, Natalia C. “(M, N )-coherent pairs of order (m, k) and Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 256 (2014), pp. 16–35. de Jesus, Marcio Dinis do Nascimento and Petronilho, Jose Carlos. “On linearly related sequences of derivatives of orthogonal polynomials”. In: J. Math. Anal. Appl. 347.2 (2008), pp. 482–492. de Jesus, Marcio Dinis do Nascimento and Petronilho, Jose Carlos. “Sobolev orthogonal polynomials and (M, N )-coherent pairs of measures”. In: J. Comput. Appl. Math. 237.1 (2013), pp. 83–101. Delgado, Antonia M., Fernandez, Lidia, Lubinsky, Doron S., Perez, Teresa E., and Piñar, Miguel A. “Sobolev orthogonal polynomials on the unit ball via outward normal derivatives”. In: J. Math. Anal. Appl. 440.2 (2016), pp. 716– 740. Delgado, Antonia M. and Marcellan, Francisco. “Companion linear functionals and Sobolev inner products: A case study”. In: Methods Appl. Anal. 11.2 (2004), pp. 237–266. Delgado, Antonia M., Perez, Teresa E., and Piñar, Miguel A. “Sobolev-type orthogonal polynomials on the unit ball”. In: J. Approx. Theory 170 (2013), pp. 94–106. Dimitrov, Dimitar K., Marcellan, Francisco, and Rafaeli, Fernando R. “Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials”. In: J. Math. Anal. Appl. 368.1 (2010), pp. 80–89. Dimitrov, Dimitar K., Mello, Mirela V., and Rafaeli, Fernando R. “Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials”. In: Appl. Numer. Math. 60.3 (2010), pp. 263–276. Dueñas, Herbert A. and Garza, Luis E. “Jacobi-Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation - a non-diagonal case”. In: Integral Transforms Spec. Funct. 24.1 (2013), pp. 70–83. Dueñas, Herbert A., Garza, Luis E., and Piñar, Miguel A. “A higher order Sobolev-type inner product for orthogonal polynomials in several variables”. In: Numer. Algorithms 68.1 (2015), pp. 35–46. Dueñas, Herbert A., Huertas, Edmundo J., and Marcellan, Francisco. “Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials”. In: Numer. Algorithms 60.1 (2012), pp. 51–73. Dueñas, Herbert A. and Marcellan, Francisco. “The Laguerre-Sobolev-type orthogonal polynomials”. In: J. Approx. Theory 162.2 (2010), pp. 421–440. Dueñas, Herbert A. and Marcellan, Francisco. “The holonomic equation of the Laguerre-Sobolev-type orthogonal polynomials: a non-diagonal case”. In: J. Difference Equ. Appl. 17.6 (2011), pp. 877–887. Dueñas, Herbert A. and Marcellan, Francisco. “The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation”. In: Rocky Mountain J. Math. 41.1 (2011), pp. 95–131. Dueñas, Herbert A., Pinzon-Cortes, Natalia, and Salazar-Morales, Omar. “Sobolev orthogonal polynomials of high order in two variables defined on product domains”. In: Integral Transforms Spec. Funct. 28.12 (Oct. 2017), pp. 988– 1008. Dueñas, Herbert A., Salazar-Morales, Omar, and Piñar, Miguel A. “Sobolev orthogonal polynomials of several variables on product domains”. In: Mediterr. J. Math. 18.5 (2021). Article 227, pp. 1–21. Duistermaat, Johannes Jisse and Kolk, Johan A. C. Distributions: theory and applications. Cornerstones. Birkhauser Basel, 2010. Dunkl, Charles F. and Xu, Yuan. Orthogonal polynomials of several variables. 2nd ed. Vol. 155. Encyclopedia of mathematics and its applications. Cambridge, United Kingdom: Cambridge University Press, 2014. Duran, Antonio J. “A generalization of Favard’s theorem for polynomials satisfying a recurrence relation”. In: J. Approx. Theory 74.1 (1993), pp. 83– 109. Evans, W. D., Littlejohn, Lance L., Marcellan, Francisco, Markett, Clemens, and Ronveaux, Andre. “On recurrence relations for Sobolev orthogonal polynomials”. In: SIAM J. Math. Anal. 26.2 (1995), pp. 446–467. Everitt, William Norrie, Kwon, Kil Hyun, Littlejohn, Lance L., and Wellman, Richard. “On the spectral analysis of the Laguerre polynomials L_n^{-k}(x) for positive integers k”. In: Spectral theory and computational methods of Sturm-Liouville problems (Knoxville, TN, 1996). Ed. by Hinton, Don and Schaefer, Philip W. Vol. 191. Lect. Notes Pure Appl. Math. New York: Marcel Dekker, Inc., 1997, pp. 251–283. Everitt, William Norrie, Littlejohn, Lance L., and Wellman, Richard. “The Sobolev orthogonality and spectral analysis of the Laguerre polynomials L_n^{-k} for positive integers k”. In: J. Comput. Appl. Math. 171.1–2 (Oct. 2004), pp. 199–234. Fernandez, Lidia, Marcellan, Francisco, Perez, Teresa E., Piñar, Miguel A., and Xu, Yuan. “Sobolev orthogonal polynomials on product domains”. In: J. Comput. Appl. Math. 284 (2015), pp. 202–215. Fernandez, Lidia, Perez, Teresa E., and Piñar, Miguel A. “Orthogonal polynomials in two variables as solutions of higher order partial differential equations”. In: J. Approx. Theory 163 (2011), pp. 84–97. Grobner, Wolfgang. “Orthogonale Polynomsysteme, die gleichzeitig mit f (x) auch deren Ableitung f′(x) approximieren”. In: Funktionalanalysis, Approximationstheorie, Numerische Mathematik. Vol. 7. International series of numerical mathematics. Springer Basel AG, 1967, pp. 24–32. Guo, Ben-Yu, Shen, Jie, and Wang, Li-Lian. “Generalized Jacobi polynomials/functions and their applications”. In: Appl. Numer. Math. 59.5 (2009), pp. 1011–1028. Horn, Roger A. and Johnson, Charles R. Matrix analysis. Cambridge University Press, 1985. Horn, Roger A. and Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, 1991. Huertas, Edmundo J., Marcellan, Francisco, and Rafaeli, Fernando R. “Zeros of orthogonal polynomials generated by canonical perturbations of measures”. In: Appl. Math. Comput. 218.13 (2012), pp. 7109–7127. Iserles, Arieh, Koch, Per Erik, Norsett, Syvert Paul, and Sanz-Serna, Jesus Marıa. “On polynomials orthogonal with respect to certain Sobolev inner products”. In: J. Approx. Theory 65.2 (1991), pp. 151–175. Jung, Il Hyo, Kwon, Kil Hyun, and Lee, Jeong Keun. “Sobolev orthogonal polynomials relative to λp(c)q(c)+ <τ, p′(x)q′(x)> ”. In: Commun. Korean Math. Soc. 12.3 (1997), pp. 603–617. Kim, D. H., Kwon, Kil Hyun, Marcellan, Francisco, and Yoon, Gang Joon. “Sobolev orthogonality and coherent pairs of moment functionals: An inverse problem”. In: Int. Math. J. 2.9 (2002), pp. 877–888. Kim, D. H., Kwon, Kil Hyun, Marcellan, Francisco, and Yoon, Gang Joon. “Zeros of Jacobi-Sobolev orthogonal polynomials”. In: Int. Math. J. 4.5 (2003), pp. 413–422. Koekoek, J. and Koekoek, Roelof. “Differential equations for generalized Jacobi polynomials”. In: J. Comput. Appl. Math. 126.1–2 (2000), pp. 1–31. Koekoek, J., Koekoek, Roelof, and Bavinck, Herman. “On differential equations for Sobolev-type Laguerre polynomials”. In: Trans. Amer. Math. Soc. 350.1 (1998), pp. 347–393. Koekoek, Roelof. “Generalizations of Laguerre polynomials”. In: J. Math. Anal. Appl. 153.2 (1990), pp. 576–590. Koekoek, Roelof. “The search for differential equations for certain sets of orthogonal polynomials”. In: J. Comput. Appl. Math. 49 (1993), pp. 111–119. Koekoek, Roelof and Meijer, Hendrik Gerrit. “A generalization of Laguerre polynomials”. In: SIAM J. Math. Anal. 24.3 (1993), pp. 768–782. Koornwinder, Tom H. “Orthogonal polynomials with weight function (1-x)^a(1 + x)^b + Mδ(x + 1) + Nδ(x-1)”. In: Canad. Math. Bull. 27.2 (1984), pp. 205–214. Krall, Allan M. “Orthogonal polynomials satisfying fourth order differential equations”. In: Proc. Roy. Soc. Edinburgh Sect. A 87.3–4 (1981), pp. 271–288. Krall, Harry LeVern. “On orthogonal polynomials satisfying a certain fourth order differential equation”. In: Pennsylvania State College Studies 1940.6 (1940), pp. 1–24. Krall, Harry LeVern and Sheffer, Isador Mitchell. “Orthogonal polynomials in two variables”. In: Ann. Mat. Pura Appl. 76.4 (1967), pp. 325–376. Kwon, Kil Hyun, Lee, J. H., and Marcellan, Francisco. “Generalized coherent pairs”. In: J. Math. Anal. Appl. 253.2 (2001), pp. 482–514. Kwon, Kil Hyun and Littlejohn, Lance L. “The orthogonality of the Laguerre polynomials {L_n^{−k}(x)} for positive integers k”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 289–303. Kwon, Kil Hyun and Littlejohn, Lance L. “Sobolev orthogonal polynomials and second-order differential equations”. In: Rocky Mountain J. Math. 28.2 (1998), pp. 547–594. Lee, Jeong Keun and Littlejohn, Lance L. “Sobolev orthogonal polynomials in two variables and second order partial differential equations”. In: J. Math. Anal. Appl. 322.2 (2006), pp. 1001–1017. Lewis, Daniel C. “Polynomial least square approximations”. In: Amer. J. Math. 69.2 (Apr. 1947), pp. 273–278. Li, Huiyuan and Shen, Jie. “Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle”. In: Math. Comp. 79.271 (2010), pp. 1621–1646. Li, Huiyuan and Xu, Yuan. “Spectral approximation on the unit ball”. In: SIAM J. Numer. Anal. 52.6 (2014), pp. 2647–2675. Marcellan, Francisco, Branquinho, Amilcar, and Petronilho, Jose Carlos. “Classical orthogonal polynomials: a functional approach”. In: Acta Appl. Math. 34.3 (1994), pp. 283–303. Marcellan, Francisco, Martınez-Finkelshtein, Andrei, and Moreno-Balcazar, Juan Jose. “k-coherence of measures with non-classical weights”. In: Margarita mathematica en memoria de Jose Javier (Chicho) Guadalupe Hernandez. Ed. by Español, Luis and Varona, Juan L. Universidad de La Rioja. Logroño, Spain: Servicio de Publicaciones, 2001, pp. 77–83. Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Gegenbauer-Sobolev Orthogonal Polynomials”. In: Nonlinear Numerical Methods and Rational Approximation II. Ed. by Cuyt, Annie. Vol. 296. Mathematics and Its Applications. Dordrecht: Springer, 1994, pp. 71–82. Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 93–122. Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Laguerre-Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 71.2 (1996), pp. 245–265. Marcellan, Francisco and Petronilho, Jose Carlos. “Orthogonal polynomials and coherent pairs: the classical case”. In: Indag. Math. (N.S.) 6.3 (1995), pp. 287–307. Marcellan, Francisco and Pinzon-Cortes, Natalia C. “Higher order coherent pairs”. In: Acta Appl. Math. 121 (2012), pp. 105–135. Marcellan, Francisco and Pinzon-Cortes, Natalia C. “Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials”. In: Publ. Inst. Math. (Beograd) (N.S.) 96.110 (2014), pp. 193–210. Marcellan, Francisco and Rafaeli, Fernando R. “Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives”. In: Proc. Amer. Math. Soc. Vol. 139. 11. Nov. 2011, pp. 3929– 3936. Marcellan, Francisco and Ronveaux, Andre. “On a class of polynomials orthogonal with respect to a discrete Sobolev inner product”. In: Indag. Math. (N.S.) 1.4 (1990), pp. 451–464. Marcellan, Francisco and Xu, Yuan. “On Sobolev orthogonal polynomials”. In: Expo. Math. 33 (2015), pp. 308–352. Martınez, Clotilde and Piñar, Miguel A. “Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations”. In: SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016). Paper No. 020, pp. 1–11. Martınez-Finkelshtein, Andrei. “Asymptotic properties of Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 99 (1998), pp. 491–510. Martınez-Finkelshtein, Andrei. “Analytic aspects of Sobolev orthogonal polynomials revisited”. In: J. Comput. Appl. Math. 127 (2001), pp. 255–266. Meijer, Hendrik Gerrit. “Coherent pairs and zeros of Sobolev-type orthogonal polynomials”. In: Indag. Math. (N.S.) 4.2 (1993), pp. 163–176. Meijer, Hendrik Gerrit. “Sobolev orthogonal polynomials with a small number of real zeros”. In: J. Approx. Theory 77.3 (1994), pp. 305–313. Meijer, Hendrik Gerrit. “A short history of orthogonal polynomials in a Sobolev space. I. The non-discrete case”. In: 31st Dutch Mathematical Conference. Vol. 14. 1. Groningen, 1995: Nieuw Arch. Wiskd. (4), 1996, pp. 93–112. Meijer, Hendrik Gerrit. “Determination of all coherent pairs”. In: J. Approx. Theory 89.3 (1997), pp. 321–343. Mello, Mirela V., Paschoa, Vanessa G., Perez, Teresa E., and Piñar, Miguel A. “Multivariate Sobolev-type orthogonal polynomials”. In: Jaen Journal on Approximation 3.2 (2011), pp. 241–259. Molano-Molano, Luis Alejandro. “On asymptotic properties of Laguerre-Sobolev type orthogonal polynomials”. In: Arab J. Math. Sci. 19.2 (2013), pp. 173–186. Molano-Molano, Luis Alejandro. “On Laguerre-Sobolev type orthogonal polynomials: zeros and electrostatic interpretation”. In: ANZIAM J. 55.1 (2013), pp. 39–54. Perez, Teresa E. and Piñar, Miguel A. “On Sobolev orthogonality for the generalized Laguerre polynomials”. In: J. Approx. Theory 86.3 (Sept. 1996), pp. 278–285. Perez, Teresa E., Piñar, Miguel A., and Xu, Yuan. “Weighted Sobolev orthogonal polynomials on the unit ball”. In: J. Approx. Theory 171 (2013), pp. 84–104. Pinar, Miguel A. and Xu, Yuan. “Orthogonal polynomials and partial differential equations on the unit ball”. In: Proc. Amer. Math. Soc. 137.9 (Sept. 2009), pp. 2979–2987. Saint Raymond, Xavier. Elementary introduction to the theory of pseudodifferential operators. Studies in advanced mathematics. CRC Press, Inc., 1991. Salazar-Morales, Omar and Dueñas, Herbert A. “Laguerre-Gegenbauer-Sobolev orthogonal polynomials in two variables on product domains”. In: Rev. Colombiana Mat. (2021). Accepted. To appear. Salazar-Morales, Omar and Dueñas, Herbert A. “Partial differential equations for some families of Sobolev orthogonal polynomials”. Submitted to journal. 2022. Schafke, Friedrich Wilhelm. “Zu den Orthogonalpolynomen von Althammer”. In: J. Reine Angew. Math. 252 (1972), pp. 195–199. Schafke, Friedrich Wilhelm and Wolf, Gerhard. “Einfache verallgemeinerte klassische Orthogonalpolynome”. In: J. Reine Angew. Math. 262/263 (1973), pp. 339–355. Sharapudinov, Idris Idrisovich. “Sobolev orthogonal polynomials associated with Chebyshev polynomials of the first kind and the Cauchy problem for ordinary differential equations”. In: Differ. Equ. 54.12 (2018), pp. 1602–1619. Sharapudinov, Idris Idrisovich. “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”. In: Sb. Math. 209.9 (2018), pp. 1390–1417. Sharapudinov, Idris Idrisovich. “Sobolev-orthogonal systems of functions associated with an orthogonal system”. In: Izv. Math. 82.1 (2018), pp. 212– 244. Sharapudinov, Idris Idrisovich. “Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs”. In: Izv. Math. 83.2 (2019), pp. 391–412. Shen, Jie, Wang, Li-Lian, and Li, Huiyuan. “A triangular spectral element method using fully tensorial rational basis functions”. In: SIAM J. Numer. Anal. 47.3 (2009), pp. 1619–1650. Suetin, P. K. Orthogonal polynomials in two variables. Vol. 3. Analytical Methods and Special Functions. Gordon and Breach Science Publishers, 1988. Szego, Gabor. Orthogonal polynomials. 4th ed. Vol. 23. Amer. Math. Soc. Colloq. Publ. Providence, Rhode Island: American Mathematical Society, 1975. Xu, Yuan. “A family of Sobolev orthogonal polynomials on the unit ball”. In: J. Approx. Theory 138 (2006), pp. 232–241. Xu, Yuan. “Sobolev orthogonal polynomials defined via gradient on the unit ball”. In: J. Approx. Theory 152 (2008), pp. 52–65. Xu, Yuan. “Approximation and orthogonality in Sobolev spaces on a triangle”. In: Constr. Approx. 46.2 (2017), pp. 349–434. Xu, Yuan. “Orthogonal polynomials and Fourier orthogonal series on a cone”. In: J. Fourier Anal. Appl. 26.3 (2020). Paper 36, pp. 1–42 |
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Universidad Nacional de Colombia |
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Facultad de Ciencias |
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Bogotá, Colombia |
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Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Dueñas Ruiz, Herbert Alonsodc9037926aef6e1d62e8b9089c4c098cSalazar Morales, Omar2fc777c4cda39e88b11c855356fbe2aaGrupo de Investigación en Polinomios Ortogonales y Aplicaciones2022-08-09T21:04:55Z2022-08-09T21:04:55Z2022-02-01https://repositorio.unal.edu.co/handle/unal/81829Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/Texto, ecuaciones, fórmulasformúlas matemáticasEn este trabajo estudiamos algunas propiedades algebraicas y analíticas de los polinomios ortogonales en varias variables reales con respecto a un producto interno Sobolev continuo-discreto. Consideramos los polinomios Sobolev sobre diferentes dominios, a saber: un dominio producto; la bola unitaria; el simplex; y el cono. Nuestros principales resultados consisten en un método iterativo de construcción de los polinomios ortogonales con respecto al producto interno, propiedades que involucran su parte principal (continua), una fórmula de conexión, y algunos resultados sobre ecuaciones diferenciales parciales. Con el fin de ilustrar nuestras principales ideas, al final de este trabajo presentamos varios ejemplos numéricos en dos variables. Además, discutimos algunos problemas abiertos.In this work we study some algebraic and analytical properties of the orthogonal polynomials in several real variables with respect to a continuous-discrete Sobolev inner product. We consider the Sobolev polynomials on different domains, namely: a product domain; the unit ball; the simplex; and the cone. Our main results consist of an iterative method for constructing the orthogonal polynomials, properties that involve the main (continuous) part of this inner product, a connection formula, and some results on partial differential equations. In order to illustrate our main ideas, at the end of this work we present some numerical examples in two variables. In addition, we discuss some open problems.Facultad de Ciencias, Sede BogotáDoctoradoDoctor en Ciencias - MatemáticasPolinomios ortogonales en varias variablesvii, 137 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::515 - AnálisisPolinomios ortogonalesPolinomios SobolevPolinomios en varias variablesProductos internosProductos internos SobolevEcuaciones diferencialesEcuaciones diferenciales parcialesOrthogonal polynomialsSobolev polynomialsPolynomials in several variablesInner productsSobolev inner productsDifferential equationsPartial differential equationsPropiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variablesProperties of some Sobolev orthogonal polynomial systems in several variablesTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDAbramowitz, Milton and Stegun, Irene Ann, eds. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Applied mathematics series - 55. Washington, D.C.: National Bureau of Standards, June 1964.Aktas, Rabia and Xu, Yuan. “Sobolev orthogonal polynomials on a simplex”. In: Int. Math. Res. Not. IMRN 2013.13 (2012), pp. 3087–3131.Alfaro, Manuel, Alvarez de Morales, Maria, and Rezola, M. Luisa. “Orthogonality of the Jacobi polynomials with negative integer parameters”. In: J. Comput. Appl. Math. 145.2 (2002), pp. 379–386.Alfaro, Manuel, Lopez, Guillermo, and Rezola, M. Luisa. “Some properties of zeros of Sobolev-type orthogonal polynomials”. In: J. Comput. Appl. Math. 69.1 (1996), pp. 171–179.Alfaro, Manuel, Marcellan, Francisco, Peña, Ana, and Rezola, M. Luisa. “On linearly related orthogonal polynomials and their functionals”. In: J. Math. Anal. Appl. 287.1 (2003), pp. 307–319.Alfaro, Manuel, Perez, Teresa E., Piñar, Miguel A., and Rezola, M. Luisa. “Sobolev orthogonal polynomials: the discrete-continuous case”. In: Methods Appl. Anal. 6.4 (1999), pp. 593–616.Althammer, Peter. “Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation”. In: J. Reine Angew. Math. 211 (Apr. 1962), pp. 192–204.Alvarez de Morales, Maria, Perez, Teresa E., and Piñar, Miguel A. “Sobolev orthogonality for the Gegenbauer polynomials {C_n^{−N+1/2}}n≥0”. In: J. Comput. Appl. Math. 100.1 (1998), pp. 111–120.Alvarez-Nodarse, Renato. Polinomios hipergeométricos clásicos y q-polinomios. Monografías del seminario matemático García de Galdeano. Prensas Universitarias de Zaragoza. Departamento de Matemáticas, Universidad de Zaragoza, 2003.Alvarez-Nodarse, Renato and Marcellan, Francisco. “A generalization of the class Laguerre polynomials: asymptotic properties and zeros”. In: Appl. Anal. 62.3–4 (1996), pp. 349–366.Alvarez-Nodarse, Renato and Moreno-Balcazar, Juan Jose. “Asymptotic properties of generalized Laguerre orthogonal polynomials”. In: Indag. Math. (N.S.) 15.2 (2004), pp. 151–165.Arvesu, Jorge, Alvarez-Nodarse, Renato, Marcellan, Francisco, and Pan, Ke-Lin. “Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros”. In: J. Comput. Appl. Math. 90.2 (1998), pp. 135–156.Atkinson, Kendall and Hansen, Olaf. “Solving the nonlinear Poisson equation on the unit disk”. In: J. Integral Equations Appl. 17.3 (2005), pp. 223–241.Bavinck, Herman and Meijer, Hendrik Gerrit. “Orthogonal polynomials with respect to a symmetric inner product involving derivatives”. In: Appl. Anal. 33.1–2 (1989), pp. 103–117.Bavinck, Herman and Meijer, Hendrik Gerrit. “On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations”. In: Indag. Math. (N.S.) 1.1 (1990), pp. 7–14.Berti, Andrea C., Bracciali, Cleonice F., and Sri Ranga, Alagacone. “Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials”. In: Numer. Algorithms 34 (2003), pp. 203–216.Berti, Andrea C. and Sri Ranga, Alagacone. “Companion orthogonal polynomials: some applications”. In: Appl. Numer. Math. 39.2 (2001), pp. 127– 149.Bracciali, Cleonice F., Delgado, Antonia M., Fernandez, Lidia, Perez, Teresa E., and Piñar, Miguel A. “New steps on Sobolev orthogonality in two variables”. In: J. Comput. Appl. Math. 235.4 (2010), pp. 916–926.Brenner, J. “Über eine Erweiterung des Orthogonaltätsbegriffes bei Polynomen”. In: Proceedings of the Conference on Constructive Theory of Functions. Ed. by Alexits, G. and Stechkin, S. B. Budapest: Akademiai Kiado, Aug. 1972, pp. 77–83.Burris, Stanley and Sankappanavar, Hanamantagouda P. A Course in Universal Algebra. Vol. 78. Graduate Texts in Mathematics. New York: Springer- Verlag, 1981.Chihara, Theodore S. An introduction to orthogonal polynomials. Gordon and Breach, Science Publishers, Inc., 1978.Cohen, Edgar A. “Zero distribution and behavior of orthogonal polynomials in the Sobolev space W^{1,2}[-1,1]”. In: SIAM J. Math. Anal. 6.1 (1975), pp. 105– 116.Dai, Feng and Xu, Yuan. “Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball”. In: J. Approx. Theory 163.10 (2011), pp. 1400– 1418.Dai, Feng and Xu, Yuan. Approximation theory and harmonic analysis on spheres and balls. Springer monographs in mathematics. New York: Springer, 2013.De Bruin, Marcel G. “A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces”. In: J. Comput. Appl. Math. 49 (1993), pp. 27– 35.De Bruin, Marcel G. and Meijer, Hendrik Gerrit. “Zeros of orthogonal polynomials in a non-discrete Sobolev space”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 233–246.de Jesus, Marcio Dinis do Nascimento, Marcellan, Francisco, Petronilho, Jose Carlos, and Pinzon-Cortes, Natalia C. “(M, N )-coherent pairs of order (m, k) and Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 256 (2014), pp. 16–35.de Jesus, Marcio Dinis do Nascimento and Petronilho, Jose Carlos. “On linearly related sequences of derivatives of orthogonal polynomials”. In: J. Math. Anal. Appl. 347.2 (2008), pp. 482–492.de Jesus, Marcio Dinis do Nascimento and Petronilho, Jose Carlos. “Sobolev orthogonal polynomials and (M, N )-coherent pairs of measures”. In: J. Comput. Appl. Math. 237.1 (2013), pp. 83–101.Delgado, Antonia M., Fernandez, Lidia, Lubinsky, Doron S., Perez, Teresa E., and Piñar, Miguel A. “Sobolev orthogonal polynomials on the unit ball via outward normal derivatives”. In: J. Math. Anal. Appl. 440.2 (2016), pp. 716– 740.Delgado, Antonia M. and Marcellan, Francisco. “Companion linear functionals and Sobolev inner products: A case study”. In: Methods Appl. Anal. 11.2 (2004), pp. 237–266.Delgado, Antonia M., Perez, Teresa E., and Piñar, Miguel A. “Sobolev-type orthogonal polynomials on the unit ball”. In: J. Approx. Theory 170 (2013), pp. 94–106.Dimitrov, Dimitar K., Marcellan, Francisco, and Rafaeli, Fernando R. “Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials”. In: J. Math. Anal. Appl. 368.1 (2010), pp. 80–89.Dimitrov, Dimitar K., Mello, Mirela V., and Rafaeli, Fernando R. “Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials”. In: Appl. Numer. Math. 60.3 (2010), pp. 263–276.Dueñas, Herbert A. and Garza, Luis E. “Jacobi-Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation - a non-diagonal case”. In: Integral Transforms Spec. Funct. 24.1 (2013), pp. 70–83.Dueñas, Herbert A., Garza, Luis E., and Piñar, Miguel A. “A higher order Sobolev-type inner product for orthogonal polynomials in several variables”. In: Numer. Algorithms 68.1 (2015), pp. 35–46.Dueñas, Herbert A., Huertas, Edmundo J., and Marcellan, Francisco. “Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials”. In: Numer. Algorithms 60.1 (2012), pp. 51–73.Dueñas, Herbert A. and Marcellan, Francisco. “The Laguerre-Sobolev-type orthogonal polynomials”. In: J. Approx. Theory 162.2 (2010), pp. 421–440.Dueñas, Herbert A. and Marcellan, Francisco. “The holonomic equation of the Laguerre-Sobolev-type orthogonal polynomials: a non-diagonal case”. In: J. Difference Equ. Appl. 17.6 (2011), pp. 877–887.Dueñas, Herbert A. and Marcellan, Francisco. “The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation”. In: Rocky Mountain J. Math. 41.1 (2011), pp. 95–131.Dueñas, Herbert A., Pinzon-Cortes, Natalia, and Salazar-Morales, Omar. “Sobolev orthogonal polynomials of high order in two variables defined on product domains”. In: Integral Transforms Spec. Funct. 28.12 (Oct. 2017), pp. 988– 1008.Dueñas, Herbert A., Salazar-Morales, Omar, and Piñar, Miguel A. “Sobolev orthogonal polynomials of several variables on product domains”. In: Mediterr. J. Math. 18.5 (2021). Article 227, pp. 1–21.Duistermaat, Johannes Jisse and Kolk, Johan A. C. Distributions: theory and applications. Cornerstones. Birkhauser Basel, 2010.Dunkl, Charles F. and Xu, Yuan. Orthogonal polynomials of several variables. 2nd ed. Vol. 155. Encyclopedia of mathematics and its applications. Cambridge, United Kingdom: Cambridge University Press, 2014.Duran, Antonio J. “A generalization of Favard’s theorem for polynomials satisfying a recurrence relation”. In: J. Approx. Theory 74.1 (1993), pp. 83– 109.Evans, W. D., Littlejohn, Lance L., Marcellan, Francisco, Markett, Clemens, and Ronveaux, Andre. “On recurrence relations for Sobolev orthogonal polynomials”. In: SIAM J. Math. Anal. 26.2 (1995), pp. 446–467.Everitt, William Norrie, Kwon, Kil Hyun, Littlejohn, Lance L., and Wellman, Richard. “On the spectral analysis of the Laguerre polynomials L_n^{-k}(x) for positive integers k”. In: Spectral theory and computational methods of Sturm-Liouville problems (Knoxville, TN, 1996). Ed. by Hinton, Don and Schaefer, Philip W. Vol. 191. Lect. Notes Pure Appl. Math. New York: Marcel Dekker, Inc., 1997, pp. 251–283.Everitt, William Norrie, Littlejohn, Lance L., and Wellman, Richard. “The Sobolev orthogonality and spectral analysis of the Laguerre polynomials L_n^{-k} for positive integers k”. In: J. Comput. Appl. Math. 171.1–2 (Oct. 2004), pp. 199–234.Fernandez, Lidia, Marcellan, Francisco, Perez, Teresa E., Piñar, Miguel A., and Xu, Yuan. “Sobolev orthogonal polynomials on product domains”. In: J. Comput. Appl. Math. 284 (2015), pp. 202–215.Fernandez, Lidia, Perez, Teresa E., and Piñar, Miguel A. “Orthogonal polynomials in two variables as solutions of higher order partial differential equations”. In: J. Approx. Theory 163 (2011), pp. 84–97.Grobner, Wolfgang. “Orthogonale Polynomsysteme, die gleichzeitig mit f (x) auch deren Ableitung f′(x) approximieren”. In: Funktionalanalysis, Approximationstheorie, Numerische Mathematik. Vol. 7. International series of numerical mathematics. Springer Basel AG, 1967, pp. 24–32.Guo, Ben-Yu, Shen, Jie, and Wang, Li-Lian. “Generalized Jacobi polynomials/functions and their applications”. In: Appl. Numer. Math. 59.5 (2009), pp. 1011–1028.Horn, Roger A. and Johnson, Charles R. Matrix analysis. Cambridge University Press, 1985.Horn, Roger A. and Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, 1991.Huertas, Edmundo J., Marcellan, Francisco, and Rafaeli, Fernando R. “Zeros of orthogonal polynomials generated by canonical perturbations of measures”. In: Appl. Math. Comput. 218.13 (2012), pp. 7109–7127.Iserles, Arieh, Koch, Per Erik, Norsett, Syvert Paul, and Sanz-Serna, Jesus Marıa. “On polynomials orthogonal with respect to certain Sobolev inner products”. In: J. Approx. Theory 65.2 (1991), pp. 151–175.Jung, Il Hyo, Kwon, Kil Hyun, and Lee, Jeong Keun. “Sobolev orthogonal polynomials relative to λp(c)q(c)+ <τ, p′(x)q′(x)> ”. In: Commun. Korean Math. Soc. 12.3 (1997), pp. 603–617.Kim, D. H., Kwon, Kil Hyun, Marcellan, Francisco, and Yoon, Gang Joon. “Sobolev orthogonality and coherent pairs of moment functionals: An inverse problem”. In: Int. Math. J. 2.9 (2002), pp. 877–888.Kim, D. H., Kwon, Kil Hyun, Marcellan, Francisco, and Yoon, Gang Joon. “Zeros of Jacobi-Sobolev orthogonal polynomials”. In: Int. Math. J. 4.5 (2003), pp. 413–422.Koekoek, J. and Koekoek, Roelof. “Differential equations for generalized Jacobi polynomials”. In: J. Comput. Appl. Math. 126.1–2 (2000), pp. 1–31.Koekoek, J., Koekoek, Roelof, and Bavinck, Herman. “On differential equations for Sobolev-type Laguerre polynomials”. In: Trans. Amer. Math. Soc. 350.1 (1998), pp. 347–393.Koekoek, Roelof. “Generalizations of Laguerre polynomials”. In: J. Math. Anal. Appl. 153.2 (1990), pp. 576–590.Koekoek, Roelof. “The search for differential equations for certain sets of orthogonal polynomials”. In: J. Comput. Appl. Math. 49 (1993), pp. 111–119.Koekoek, Roelof and Meijer, Hendrik Gerrit. “A generalization of Laguerre polynomials”. In: SIAM J. Math. Anal. 24.3 (1993), pp. 768–782.Koornwinder, Tom H. “Orthogonal polynomials with weight function (1-x)^a(1 + x)^b + Mδ(x + 1) + Nδ(x-1)”. In: Canad. Math. Bull. 27.2 (1984), pp. 205–214.Krall, Allan M. “Orthogonal polynomials satisfying fourth order differential equations”. In: Proc. Roy. Soc. Edinburgh Sect. A 87.3–4 (1981), pp. 271–288.Krall, Harry LeVern. “On orthogonal polynomials satisfying a certain fourth order differential equation”. In: Pennsylvania State College Studies 1940.6 (1940), pp. 1–24.Krall, Harry LeVern and Sheffer, Isador Mitchell. “Orthogonal polynomials in two variables”. In: Ann. Mat. Pura Appl. 76.4 (1967), pp. 325–376.Kwon, Kil Hyun, Lee, J. H., and Marcellan, Francisco. “Generalized coherent pairs”. In: J. Math. Anal. Appl. 253.2 (2001), pp. 482–514.Kwon, Kil Hyun and Littlejohn, Lance L. “The orthogonality of the Laguerre polynomials {L_n^{−k}(x)} for positive integers k”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 289–303.Kwon, Kil Hyun and Littlejohn, Lance L. “Sobolev orthogonal polynomials and second-order differential equations”. In: Rocky Mountain J. Math. 28.2 (1998), pp. 547–594.Lee, Jeong Keun and Littlejohn, Lance L. “Sobolev orthogonal polynomials in two variables and second order partial differential equations”. In: J. Math. Anal. Appl. 322.2 (2006), pp. 1001–1017.Lewis, Daniel C. “Polynomial least square approximations”. In: Amer. J. Math. 69.2 (Apr. 1947), pp. 273–278.Li, Huiyuan and Shen, Jie. “Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle”. In: Math. Comp. 79.271 (2010), pp. 1621–1646.Li, Huiyuan and Xu, Yuan. “Spectral approximation on the unit ball”. In: SIAM J. Numer. Anal. 52.6 (2014), pp. 2647–2675.Marcellan, Francisco, Branquinho, Amilcar, and Petronilho, Jose Carlos. “Classical orthogonal polynomials: a functional approach”. In: Acta Appl. Math. 34.3 (1994), pp. 283–303.Marcellan, Francisco, Martınez-Finkelshtein, Andrei, and Moreno-Balcazar, Juan Jose. “k-coherence of measures with non-classical weights”. In: Margarita mathematica en memoria de Jose Javier (Chicho) Guadalupe Hernandez. Ed. by Español, Luis and Varona, Juan L. Universidad de La Rioja. Logroño, Spain: Servicio de Publicaciones, 2001, pp. 77–83.Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Gegenbauer-Sobolev Orthogonal Polynomials”. In: Nonlinear Numerical Methods and Rational Approximation II. Ed. by Cuyt, Annie. Vol. 296. Mathematics and Its Applications. Dordrecht: Springer, 1994, pp. 71–82.Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case”. In: Special functions (Torino, 1993). Vol. 2. Ann. Numer. Math. 1–4. 1995, pp. 93–122.Marcellan, Francisco, Perez, Teresa E., and Piñar, Miguel A. “Laguerre-Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 71.2 (1996), pp. 245–265.Marcellan, Francisco and Petronilho, Jose Carlos. “Orthogonal polynomials and coherent pairs: the classical case”. In: Indag. Math. (N.S.) 6.3 (1995), pp. 287–307.Marcellan, Francisco and Pinzon-Cortes, Natalia C. “Higher order coherent pairs”. In: Acta Appl. Math. 121 (2012), pp. 105–135.Marcellan, Francisco and Pinzon-Cortes, Natalia C. “Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials”. In: Publ. Inst. Math. (Beograd) (N.S.) 96.110 (2014), pp. 193–210.Marcellan, Francisco and Rafaeli, Fernando R. “Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives”. In: Proc. Amer. Math. Soc. Vol. 139. 11. Nov. 2011, pp. 3929– 3936.Marcellan, Francisco and Ronveaux, Andre. “On a class of polynomials orthogonal with respect to a discrete Sobolev inner product”. In: Indag. Math. (N.S.) 1.4 (1990), pp. 451–464.Marcellan, Francisco and Xu, Yuan. “On Sobolev orthogonal polynomials”. In: Expo. Math. 33 (2015), pp. 308–352.Martınez, Clotilde and Piñar, Miguel A. “Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations”. In: SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016). Paper No. 020, pp. 1–11.Martınez-Finkelshtein, Andrei. “Asymptotic properties of Sobolev orthogonal polynomials”. In: J. Comput. Appl. Math. 99 (1998), pp. 491–510.Martınez-Finkelshtein, Andrei. “Analytic aspects of Sobolev orthogonal polynomials revisited”. In: J. Comput. Appl. Math. 127 (2001), pp. 255–266.Meijer, Hendrik Gerrit. “Coherent pairs and zeros of Sobolev-type orthogonal polynomials”. In: Indag. Math. (N.S.) 4.2 (1993), pp. 163–176.Meijer, Hendrik Gerrit. “Sobolev orthogonal polynomials with a small number of real zeros”. In: J. Approx. Theory 77.3 (1994), pp. 305–313.Meijer, Hendrik Gerrit. “A short history of orthogonal polynomials in a Sobolev space. I. The non-discrete case”. In: 31st Dutch Mathematical Conference. Vol. 14. 1. Groningen, 1995: Nieuw Arch. Wiskd. (4), 1996, pp. 93–112.Meijer, Hendrik Gerrit. “Determination of all coherent pairs”. In: J. Approx. Theory 89.3 (1997), pp. 321–343.Mello, Mirela V., Paschoa, Vanessa G., Perez, Teresa E., and Piñar, Miguel A. “Multivariate Sobolev-type orthogonal polynomials”. In: Jaen Journal on Approximation 3.2 (2011), pp. 241–259.Molano-Molano, Luis Alejandro. “On asymptotic properties of Laguerre-Sobolev type orthogonal polynomials”. In: Arab J. Math. Sci. 19.2 (2013), pp. 173–186.Molano-Molano, Luis Alejandro. “On Laguerre-Sobolev type orthogonal polynomials: zeros and electrostatic interpretation”. In: ANZIAM J. 55.1 (2013), pp. 39–54.Perez, Teresa E. and Piñar, Miguel A. “On Sobolev orthogonality for the generalized Laguerre polynomials”. In: J. Approx. Theory 86.3 (Sept. 1996), pp. 278–285.Perez, Teresa E., Piñar, Miguel A., and Xu, Yuan. “Weighted Sobolev orthogonal polynomials on the unit ball”. In: J. Approx. Theory 171 (2013), pp. 84–104.Pinar, Miguel A. and Xu, Yuan. “Orthogonal polynomials and partial differential equations on the unit ball”. In: Proc. Amer. Math. Soc. 137.9 (Sept. 2009), pp. 2979–2987.Saint Raymond, Xavier. Elementary introduction to the theory of pseudodifferential operators. Studies in advanced mathematics. CRC Press, Inc., 1991.Salazar-Morales, Omar and Dueñas, Herbert A. “Laguerre-Gegenbauer-Sobolev orthogonal polynomials in two variables on product domains”. In: Rev. Colombiana Mat. (2021). Accepted. To appear.Salazar-Morales, Omar and Dueñas, Herbert A. “Partial differential equations for some families of Sobolev orthogonal polynomials”. Submitted to journal. 2022.Schafke, Friedrich Wilhelm. “Zu den Orthogonalpolynomen von Althammer”. In: J. Reine Angew. Math. 252 (1972), pp. 195–199.Schafke, Friedrich Wilhelm and Wolf, Gerhard. “Einfache verallgemeinerte klassische Orthogonalpolynome”. In: J. Reine Angew. Math. 262/263 (1973), pp. 339–355.Sharapudinov, Idris Idrisovich. “Sobolev orthogonal polynomials associated with Chebyshev polynomials of the first kind and the Cauchy problem for ordinary differential equations”. In: Differ. Equ. 54.12 (2018), pp. 1602–1619.Sharapudinov, Idris Idrisovich. “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”. In: Sb. Math. 209.9 (2018), pp. 1390–1417.Sharapudinov, Idris Idrisovich. “Sobolev-orthogonal systems of functions associated with an orthogonal system”. In: Izv. Math. 82.1 (2018), pp. 212– 244.Sharapudinov, Idris Idrisovich. “Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs”. In: Izv. Math. 83.2 (2019), pp. 391–412.Shen, Jie, Wang, Li-Lian, and Li, Huiyuan. “A triangular spectral element method using fully tensorial rational basis functions”. In: SIAM J. Numer. Anal. 47.3 (2009), pp. 1619–1650.Suetin, P. K. Orthogonal polynomials in two variables. Vol. 3. Analytical Methods and Special Functions. Gordon and Breach Science Publishers, 1988.Szego, Gabor. Orthogonal polynomials. 4th ed. Vol. 23. Amer. Math. Soc. Colloq. Publ. Providence, Rhode Island: American Mathematical Society, 1975.Xu, Yuan. “A family of Sobolev orthogonal polynomials on the unit ball”. In: J. Approx. Theory 138 (2006), pp. 232–241.Xu, Yuan. “Sobolev orthogonal polynomials defined via gradient on the unit ball”. In: J. Approx. Theory 152 (2008), pp. 52–65.Xu, Yuan. “Approximation and orthogonality in Sobolev spaces on a triangle”. In: Constr. Approx. 46.2 (2017), pp. 349–434.Xu, Yuan. “Orthogonal polynomials and Fourier orthogonal series on a cone”. In: J. Fourier Anal. Appl. 26.3 (2020). Paper 36, pp. 1–42Sobre polinomios ortogonales en varias variables, polinomios ortogonales matriciales y pares coherentes de polinomios ortogonalesEstudiantesInvestigadoresMaestrosPadres y familiasPúblico generalORIGINAL80102001.2022.pdf80102001.2022.pdfTesis de Doctorado en Ciencias Matemáticasapplication/pdf1026232https://repositorio.unal.edu.co/bitstream/unal/81829/1/80102001.2022.pdf0d357ada4bc2e9aa01d677b344cbf6a6MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-84074https://repositorio.unal.edu.co/bitstream/unal/81829/2/license.txt8153f7789df02f0a4c9e079953658ab2MD52THUMBNAIL80102001.2022.pdf.jpg80102001.2022.pdf.jpgGenerated Thumbnailimage/jpeg5047https://repositorio.unal.edu.co/bitstream/unal/81829/3/80102001.2022.pdf.jpgfdc93db27d6e861486a69302e75d3d2eMD53unal/81829oai:repositorio.unal.edu.co:unal/818292023-08-06 23:04:08.293Repositorio Institucional Universidad Nacional de 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