Discrete-continuous Jacobi-Sobolev spaces and Fourier series

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dc.contributor.author Díaz González, Abel
dc.contributor.author Marcellán Español, Francisco José
dc.contributor.author Pijeira Cabrera, Héctor Esteban
dc.contributor.author Urbina, Wilfredo
dc.date.accessioned 2022-01-25T12:22:14Z
dc.date.available 2022-01-25T12:22:14Z
dc.date.issued 2021-03
dc.identifier.bibliographicCitation Díaz-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.
dc.identifier.issn 0126-6705
dc.identifier.uri http://hdl.handle.net/10016/33956
dc.description.abstract Let 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.
dc.description.sponsorship Authors 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.
dc.format.extent 28
dc.language.iso eng
dc.publisher Springer Nature
dc.rights © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020.
dc.subject.other Sobolev orthogonal polynomials
dc.subject.other Jacobi polynomials
dc.subject.other Fourier series
dc.title Discrete-continuous Jacobi-Sobolev spaces and Fourier series
dc.type article
dc.subject.eciencia Matemáticas
dc.identifier.doi https://doi.org/10.1007/s40840-020-00950-7
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. BES-2016-076613
dc.relation.projectID Gobierno de España. PGC2018-096504-B-C33
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 571
dc.identifier.publicationissue 2
dc.identifier.publicationlastpage 598
dc.identifier.publicationtitle Bulletin of the Malaysian Mathematical Sciences Society
dc.identifier.publicationvolume 44
dc.identifier.uxxi AR/0000026672
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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