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Discrete-continuous Jacobi-Sobolev spaces and Fourier series

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorDíaz González, Abel
dc.contributor.authorMarcellán Español, Francisco José
dc.contributor.authorPijeira Cabrera, Héctor Esteban
dc.contributor.authorUrbina, Wilfredo
dc.contributor.funderMinisterio de Economía y Competitividad (España)es
dc.date.accessioned2022-01-25T12:22:14Z
dc.date.available2022-01-25T12:22:14Z
dc.date.issued2021-03
dc.description.abstractLet p≥1,ℓ∈N,α,β>−1 and ϖ=(ω0,ω1,…,ωℓ−1)∈Rℓ. Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: ∥f∥s,p:=(∑k=0ℓ−1∣∣f(k)(ωk)∣∣p+∫1−1∣∣f(ℓ)(x)∣∣pdμα,β(x))1p, where dμα,β(x)=(1−x)α(1+x)βdx. Obviously, ∥⋅∥s,2=⟨⋅,⋅⟩s−−−−√, where ⟨⋅,⋅⟩s is the inner product ⟨f,g⟩s:=∑k=0ℓ−1f(k)(ωk)g(k)(ωk)+∫1−1f(ℓ)(x)g(ℓ)(x)dμα,β(x). In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type Lp, in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm ∥⋅∥s,p and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in ∥⋅∥s,p norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to ⟨⋅,⋅⟩s.en
dc.description.sponsorshipAuthors thank the valuable comments by the referees. Their suggestions have contributed to improve the presentation of this manuscript. The research of F. Marcellán and H. Pijeira-Cabrera was partially supported by Spanish State Research Agency, under Grant PGC2018-096504-B-C33. The research of A. Díaz-González was supported by the Research Fellowship Program, Ministry of Economy and Competitiveness of Spain, under grant BES-2016-076613.en
dc.format.extent28
dc.identifier.bibliographicCitationDíaz-González, A., Marcellán, F., Pijeira-Cabrera, H. & Urbina, W. (2020). Discrete–Continuous Jacobi–Sobolev Spaces and Fourier Series. Bulletin of the Malaysian Mathematical Sciences Society, 44(2), 571–598.en
dc.identifier.doihttps://doi.org/10.1007/s40840-020-00950-7
dc.identifier.issn0126-6705
dc.identifier.publicationfirstpage571
dc.identifier.publicationissue2
dc.identifier.publicationlastpage598
dc.identifier.publicationtitleBulletin of the Malaysian Mathematical Sciences Societyen
dc.identifier.publicationvolume44
dc.identifier.urihttps://hdl.handle.net/10016/33956
dc.identifier.uxxiAR/0000026672
dc.language.isoengen
dc.publisherSpringer Natureen
dc.relation.projectIDGobierno de España. BES-2016-076613es
dc.relation.projectIDGobierno de España. PGC2018-096504-B-C33es
dc.rights© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020.en
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.subject.otherSobolev orthogonal polynomialsen
dc.subject.otherJacobi polynomialsen
dc.subject.otherFourier seriesen
dc.titleDiscrete-continuous Jacobi-Sobolev spaces and Fourier seriesen
dc.typeresearch article*
dc.type.hasVersionAM*
dspace.entity.typePublication
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