On the behaviour of zeros of Jacobi polynomials
Autor(a) principal: | |
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Data de Publicação: | 2002 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1006/jath.2002.3671 http://hdl.handle.net/11449/21732 |
Resumo: | Denote by x(n,k)(alpha, beta) and x(n,k) (lambda) = x(n,k) (lambda - 1/2, lambda - 1/2) the zeros, in decreasing order, of the Jacobi polynomial P-n((alpha, beta))(x) and of the ultraspherical (Gegenbauer) polynomial C-n(lambda)(x), respectively. The monotonicity of x(n,k)(alpha, beta) as functions of a and beta, alpha, beta > - 1, is investigated. Necessary conditions such that the zeros of P-n((a, b)) (x) are smaller (greater) than the zeros of P-n((alpha, beta))(x) are provided. A. Markov proved that x(n,k) (a, b) < x(n,k)(α, β) (x(n,k)(a, b) > x(n,k)(alpha, beta)) for every n is an element of N and each k, 1 less than or equal to k less than or equal to n if a > alpha and b < β (a < alpha and b > beta). We prove the converse statement of Markov's theorem. The question of how large the function could be such that the products f(n)(lambda) x(n,k)(lambda), k = 1,..., [n/2] are increasing functions of lambda, for lambda > - 1/2, is also discussed. Elbert and Siafarikas proved that f(n)(lambda) = (lambda + (2n(2) + 1)/ (4n + 2))(1/2) obeys this property. We establish the sharpness of their result. (C) 2002 Elsevier B.V. (USA). |
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Repositório Institucional da UNESP |
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On the behaviour of zeros of Jacobi polynomialsDenote by x(n,k)(alpha, beta) and x(n,k) (lambda) = x(n,k) (lambda - 1/2, lambda - 1/2) the zeros, in decreasing order, of the Jacobi polynomial P-n((alpha, beta))(x) and of the ultraspherical (Gegenbauer) polynomial C-n(lambda)(x), respectively. The monotonicity of x(n,k)(alpha, beta) as functions of a and beta, alpha, beta > - 1, is investigated. Necessary conditions such that the zeros of P-n((a, b)) (x) are smaller (greater) than the zeros of P-n((alpha, beta))(x) are provided. A. Markov proved that x(n,k) (a, b) < x(n,k)(α, β) (x(n,k)(a, b) > x(n,k)(alpha, beta)) for every n is an element of N and each k, 1 less than or equal to k less than or equal to n if a > alpha and b < β (a < alpha and b > beta). We prove the converse statement of Markov's theorem. The question of how large the function could be such that the products f(n)(lambda) x(n,k)(lambda), k = 1,..., [n/2] are increasing functions of lambda, for lambda > - 1/2, is also discussed. Elbert and Siafarikas proved that f(n)(lambda) = (lambda + (2n(2) + 1)/ (4n + 2))(1/2) obeys this property. We establish the sharpness of their result. (C) 2002 Elsevier B.V. (USA).Univ Estadual Paulista, IBILCE, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, BrazilUniv Estadual Paulista, IBILCE, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, BrazilElsevier B.V.Universidade Estadual Paulista (Unesp)Dimitrov, D. K.Rodrigues, R. O.2014-05-20T14:01:35Z2014-05-20T14:01:35Z2002-06-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article224-239application/pdfhttp://dx.doi.org/10.1006/jath.2002.3671Journal of Approximation Theory. San Diego: Academic Press Inc. Elsevier B.V., v. 116, n. 2, p. 224-239, 2002.0021-9045http://hdl.handle.net/11449/2173210.1006/jath.2002.3671WOS:000176490200002WOS000176490200002.pdf1681267716971253Web of Sciencereponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengJournal of Approximation Theory0.9390,907info:eu-repo/semantics/openAccess2024-01-13T06:36:58Zoai:repositorio.unesp.br:11449/21732Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T22:53:48.032866Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
On the behaviour of zeros of Jacobi polynomials |
title |
On the behaviour of zeros of Jacobi polynomials |
spellingShingle |
On the behaviour of zeros of Jacobi polynomials Dimitrov, D. K. |
title_short |
On the behaviour of zeros of Jacobi polynomials |
title_full |
On the behaviour of zeros of Jacobi polynomials |
title_fullStr |
On the behaviour of zeros of Jacobi polynomials |
title_full_unstemmed |
On the behaviour of zeros of Jacobi polynomials |
title_sort |
On the behaviour of zeros of Jacobi polynomials |
author |
Dimitrov, D. K. |
author_facet |
Dimitrov, D. K. Rodrigues, R. O. |
author_role |
author |
author2 |
Rodrigues, R. O. |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) |
dc.contributor.author.fl_str_mv |
Dimitrov, D. K. Rodrigues, R. O. |
description |
Denote by x(n,k)(alpha, beta) and x(n,k) (lambda) = x(n,k) (lambda - 1/2, lambda - 1/2) the zeros, in decreasing order, of the Jacobi polynomial P-n((alpha, beta))(x) and of the ultraspherical (Gegenbauer) polynomial C-n(lambda)(x), respectively. The monotonicity of x(n,k)(alpha, beta) as functions of a and beta, alpha, beta > - 1, is investigated. Necessary conditions such that the zeros of P-n((a, b)) (x) are smaller (greater) than the zeros of P-n((alpha, beta))(x) are provided. A. Markov proved that x(n,k) (a, b) < x(n,k)(α, β) (x(n,k)(a, b) > x(n,k)(alpha, beta)) for every n is an element of N and each k, 1 less than or equal to k less than or equal to n if a > alpha and b < β (a < alpha and b > beta). We prove the converse statement of Markov's theorem. The question of how large the function could be such that the products f(n)(lambda) x(n,k)(lambda), k = 1,..., [n/2] are increasing functions of lambda, for lambda > - 1/2, is also discussed. Elbert and Siafarikas proved that f(n)(lambda) = (lambda + (2n(2) + 1)/ (4n + 2))(1/2) obeys this property. We establish the sharpness of their result. (C) 2002 Elsevier B.V. (USA). |
publishDate |
2002 |
dc.date.none.fl_str_mv |
2002-06-01 2014-05-20T14:01:35Z 2014-05-20T14:01:35Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1006/jath.2002.3671 Journal of Approximation Theory. San Diego: Academic Press Inc. Elsevier B.V., v. 116, n. 2, p. 224-239, 2002. 0021-9045 http://hdl.handle.net/11449/21732 10.1006/jath.2002.3671 WOS:000176490200002 WOS000176490200002.pdf 1681267716971253 |
url |
http://dx.doi.org/10.1006/jath.2002.3671 http://hdl.handle.net/11449/21732 |
identifier_str_mv |
Journal of Approximation Theory. San Diego: Academic Press Inc. Elsevier B.V., v. 116, n. 2, p. 224-239, 2002. 0021-9045 10.1006/jath.2002.3671 WOS:000176490200002 WOS000176490200002.pdf 1681267716971253 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Journal of Approximation Theory 0.939 0,907 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
224-239 application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier B.V. |
publisher.none.fl_str_mv |
Elsevier B.V. |
dc.source.none.fl_str_mv |
Web of Science reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1808129471550586880 |