# On Fourier series of a discrete Jacobi-Sobolev inner product

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 dc.contributor.author Marcellán Español, Francisco José dc.contributor.author Osilenker, Boris P. dc.contributor.author Álvarez Rocha, Ignacio dc.date.accessioned 2009-12-10T13:34:17Z dc.date.available 2009-12-10T13:34:17Z dc.date.issued 2002-07 dc.identifier.bibliographicCitation Journal of Approximation Theory, 2002, vol. 117, n. 1, p. 1-22 dc.identifier.issn 0021-9045 dc.identifier.uri http://hdl.handle.net/10016/6003 dc.description 22 pages, no figures.-- MSC2000 code: 42C05. dc.description MR#: MR1920116 (2003e:42007) dc.description Zbl#: Zbl 1019.42014 dc.description.abstract Let $\mu$ be the Jacobi measure supported on the interval $[-1,1]$ and introduce the discrete Sobolev-type inner product $$\langle f,g\rangle= \int _{-1} f(x) g(x) d\mu(x)+ \sum _{k=1} \sum N_k}_{i=0} M_{k,i} f (i)}(a_k) g (i)}(a_k),$$ where $a_k$, $1\le k\le K$, are real numbers such that $_k 1$ and $M_{k,i}> 0$ for all $k$, $i$. This paper is a continuation of [{\it F. Marcellán}, {\it B. P. Osilenker} and {\it I. A. Rocha}, "On Fourier series of Jacobi-Sobolev orthogonal polynomials", J. Inequal. Appl. 7, 673-699 (2002; Zbl 1016.42014)] and our main purpose is to study the behaviour of the Fourier series associated with such a Sobolev inner product. For an appropriate function $f$, we prove here that the Fourier-Sobolev series converges to $f$ on $(-1,1)\bigcup _{k=1}\{a_k\}$, and the derivatives of the series converge to $f (i)}(a_k)$ for all $i$ and $k$. Roughly speaking, the term appropriate means here the same as we need for a function $f$ in order to have convergence for its Fourier series associated with the standard inner product given by the measure $\mu$. No additional conditions are needed. dc.description.sponsorship The work of F. Marcellán was supported by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain BFM 2000 0206 C04 01 and by an INTAS Grant 2000/272. dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Elsevier dc.rights © Elsevier dc.subject.other Orthogonal polynomials dc.subject.other Sobolev inner product dc.subject.other Fourier series dc.title On Fourier series of a discrete Jacobi-Sobolev inner product dc.type article dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://dx.doi.org/10.1006/jath.2002.3681 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1006/jath.2002.3681 dc.rights.accessRights openAccess
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