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12 results on '"Semiclassical linear functionals"'

1. An inverse problem associated with (1,1) symmetric coherent linear functionals.

Author

Ruiz, Herbert Dueñas, Español, Francisco Marcellán, and Molano, Alejandro Molano

Subjects
*INVERSE problems, *ORTHOGONAL polynomials, *FUNCTIONALS, *SEQUENCE analysis, *POLYNOMIALS
Abstract

In this paper we study an inverse problem associated with the non-coherent algebraic relation where the sequences of monic polynomials and are orthogonal with respect to quasi-definite linear functionals u, v and w, respectively, and for We assume that v is a monic polynomial perturbation of u. Moreover, in the case we assume that u is a semiclassical linear functional of class s. In this way, we discuss the relation between the formal Stieltjes series associated with u and w. This inverse problem is motivated by the analysis of sequences of polynomials orthogonal with respect to symmetric coherent pairs of linear functionals when a symmetrization process is implemented. [ABSTRACT FROM AUTHOR]

Published
2019
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2. On the modification of non-regular linear functionals via addition of the Dirac delta function.

Author

Sghaier, Mabrouk and Khaled, Lamaa

Subjects
*POLYNOMIALS, *ORTHOGONAL polynomials, *MATRICES (Mathematics), *MATHEMATICS research, *INTEGRAL representations
Abstract

A linear functional (form) v is called regular if there exists a sequence of polynomials { S n} n≥0 with deg Sn = n which is orthogonal with respect to v. The linear functional v˜ = S1 v is not regular. We study properties of the linear functional u satisfying u = λ v˜ + δ a, where a ∈ ℂ and λ ∈ ℂ - {0}. Necessary and sufficient conditions are given for the regularity of the linear functional u. The corresponding tridiagonal matrices and associated polynomials are also studied. A study of the semiclassical character of the found families is done. We conclude by giving some examples. [ABSTRACT FROM AUTHOR]

Published
2015
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3. Higher Order Coherent Pairs.

Author

Marcellán, Francisco and Pinzón, Natalia

Subjects
*MATHEMATICAL sequences, *ORTHOGONAL polynomials, *SOBOLEV spaces, *INNER product spaces, *STIELTJES transform, *MATHEMATICAL functions
Abstract

In this paper, we study necessary and sufficient conditions for the relation where { P( x)} and { R( x)} are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $\mathcal{U},\mathcal{V}$, respectively, or associated with two positive Borel measures μ, μ supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical. [ABSTRACT FROM AUTHOR]

Published
2012
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4. On semiclassical linear functionals of class s =2: classification and integral representations.

Author

Marcellán, F., Sghaier, M., and Zaatra, M.

Subjects
*FUNCTIONALS, *INTEGRAL representations, *ORTHOGONAL polynomials, *EQUATIONS, *PERTURBATION theory
Abstract

In this paper, we obtain all the semiclassical linear functionals of class two taking into account the irreducible expression of the corresponding Pearson equation. We focus our attention in their integral representations. Thus, some linear functionals very well known in the literature, associated with perturbations of classical linear functionals, as well as new linear functionals appear which have not been studied as far as we know. [ABSTRACT FROM PUBLISHER]

Published
2012
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5. On inverse problem leading to second-order linear functionals

Author

Sfaxi, Ridha

Subjects
*INVERSE problems, *FUNCTIONAL analysis, *POLYNOMIALS, *ORTHOGONAL polynomials, *POINT mappings (Mathematics), *INTEGRAL representations, *APPROXIMATION theory
Abstract

Abstract: A linear functional is said to be positive-definite if and only if , for all non-zero polynomials with real coefficients . In this paper, we provide a new construction process of a positive-definite linear functional from positive-definite linear functional data. Indeed, for any non-zero real and any positive-definite linear functional , we show that the linear functional satisfying is also positive-definite. This process allows us to construct a second-order positive-definite linear functional from a semiclassical positive-definite linear functional. Finally, we give an illustrative example. [Copyright &y& Elsevier]

Published
2010
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6. Second structure relation for semiclassical orthogonal polynomials

Author

Marcellán, Francisco and Sfaxi, Ridha

Subjects
*ORTHOGONAL polynomials, *FOURIER analysis, *ORTHOGONAL functions, *POLYNOMIALS, *APPROXIMATION theory
Abstract

Abstract: Classical orthogonal polynomials are characterized from their orthogonality and by a first or second structure relation. For the semiclassical orthogonal polynomials (a generalization of the classical ones), we find only the first structure relation in the literature. In this paper, we establish a second structure relation. In particular, we deduce it by means of a general finite-type relation between a semiclassical polynomial sequence and the sequence of its monic derivatives. [Copyright &y& Elsevier]

Published
2007
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7. What is beyond coherent pairs of orthogonal polynomials?

Author

Teresa E. Pérez, Miguel A. Piñar, J. Petronilho, and Francisco Marcellán

Subjects
Discrete mathematics, Orthogonal polynomials, Orthogonal transformation, Applied Mathematics, Order (ring theory), Semiclassical linear functionals, Context (language use), Coherent pairs, Sobolev space, Computational Mathematics, Orthogonality, Symmetrically coherent pairs, Coherence (signal processing), Focus (optics), Regular linear functionals, Mathematics
Abstract

Usually, coherent pairs of orthogonal polynomials have been considered in the wider context of Sobolev orthogonality. In this paper, we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals. We deduce some conditions about the linear functionals in order that the corresponding orthogonal polynomial sequences constitute a coherent pair.

Published
1995
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8. On the semiclassical character of orthogonal polynomials satisfying structure relations

Author

Branquinho, A. and Rebocho, M. N.

Subjects
Structure relations, Orthogonal polynomials, Recurrence relations, Semiclassical linear functionals
Abstract

Submitted by REBOCHO (mneves@ubi.pt) on 2020-02-03T13:29:44Z No. of bitstreams: 1 preprint DMUC structure OPRL.pdf: 261134 bytes, checksum: 39720dc385b41d0897cd83a46a4d2352 (MD5) Approved for entry into archive by Pessoa (pfep@ubi.pt) on 2020-02-04T16:02:22Z (GMT) No. of bitstreams: 1 preprint DMUC structure OPRL.pdf: 261134 bytes, checksum: 39720dc385b41d0897cd83a46a4d2352 (MD5) Approved for entry into archive by Pessoa (pfep@ubi.pt) on 2020-02-04T16:04:39Z (GMT) No. of bitstreams: 1 preprint DMUC structure OPRL.pdf: 261134 bytes, checksum: 39720dc385b41d0897cd83a46a4d2352 (MD5) Made available in DSpace on 2020-02-04T16:04:39Z (GMT). No. of bitstreams: 1 preprint DMUC structure OPRL.pdf: 261134 bytes, checksum: 39720dc385b41d0897cd83a46a4d2352 (MD5) Previous issue date: 2012 info:eu-repo/semantics/publishedVersion

Published
2010

9. Continuous symmetrized Sobolev inner products of order N (II)

Author

Sanchez-Ruiz, Jorge, Marcellan, Francisco, and Bueno, M. Isabel

Subjects
Matemáticas, Sobolev inner product, Orthogonal polynomials, Semiclassical linear functionals, Recurrence relation, Symmetrization process
Abstract

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies Q2n(x)=Pn(x2), Q2n+1(x)=xRn(x2) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper we consider the particular case when all the measures that define the symmetrized Sobolev inner product are equal, absolutely continuous and semiclassical. Under such restrictions, we give explicit algebraic relations between the sequences {Pn} and {Rn}, as well as higher-order recurrence relations that they satisfy., The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grants BFM 2003-06335-C03-02 (M.I.B., F.M., J.S.R.) and BFM2001-3878-C02-01 (J.S.R.), INTAS Research Network NeCCA INTAS 03-51-6637 (F.M.), and the Junta de Andalucía research group FQM-0207 (J.S.R).

Published
2006

10. Companion linear functionals and Sobolev inner products

Author

Delgado, Antonia M. and Marcellán Español, Francisco José

Subjects
Inverse problems, Orthogonal polynomials, Matemáticas, Semiclassical linear functionals, Perturbations of linear functionals
Abstract

30 pages, no figures.-- MSC2000 codes: Primary 33C45, 42C05. MR#: MR2143522 (2006f:42022) Zbl#: Zbl 1087.42020 The present paper deals with the solution of an inverse problem in the theory of orthogonal polynomials. It was motivated by a characterization result concerning sequences of polynomials orthogonal with respect to a Sobolev inner product when they can be recursively generated in terms of orthogonal polynomial sequences associated with the measure involved in the standard component. More precisely, we obtain the set of pairs of quasi–definite linear functionals such that their corresponding sequences of monic orthogonal polynomials {Pn} and {Rn} are related by a differential expression $$ \frac{R'_{n+1}(x)}{n+1}+b_n\frac{R'_n(x)}{n}=P_n(x)+a_nP_{n-1}(x) \tag 2$$ where $ b_n\ne0$ for every n ∈ N. The work of the authors has been supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant BFM 2003–06335–C03–02 and INTAS Research Network NeCCA INTAS 03–31–6637. Publicado

Published
2004

11. Continuous symmetric Sobolev inner products

Author

Bueno Cachadiña, María Isabel and Marcellán Español, Francisco José

Subjects
Sobolev inner products, Orthogonal polynomials, Matemáticas, Recurrence relations, Semiclassical linear functionals
Abstract

24 pages, no figures.-- MSC2000 codes: 42C05. MR#: MR1953647 (2003j:42029) Zbl#: Zbl pre05368623 In this paper we consider the sequence of monic polynomials (Qn) orthogonal with respect to a symmetric Sobolev inner product. If Q_2n(x)=Pn(x^2) and Q_2n+1(x)=xRn(x^2), then we deduce the integral representation of the inner products such that (Pn) and (Rn) are, respectively, the corresponding sequences of monic orthogonal polynomials. In the semiclassical case, algebraic relations between such sequences are deduced. Finally, an application of the above results to Freud-Sobolev polynomials is given. The work of the second author was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2000-0206-C04-01 and INTAS project INTAS 2000-272. Publicado

Published
2003

12. Structure relations for orthogonal polynomials on the unit circle

Author

Amílcar Branquinho and M. N. Rebocho

Subjects
Discrete mathematics, Numerical Analysis, Polynomial, Algebra and Number Theory, Recurrence relation, Orthogonal polynomials on the unit circle, 010102 general mathematics, Recurrence relations, Structure (category theory), Semiclassical linear functionals, Semi-classical linear functionals, 010103 numerical & computational mathematics, Type (model theory), 01 natural sciences, Hermitian matrix, Structure relations, Orthogonal polynomials, Discrete Mathematics and Combinatorics, Geometry and Topology, 0101 mathematics, Complex number, Hermitian linear functionals, Mathematics
Abstract

Submitted by REBOCHO (mneves@ubi.pt) on 2020-02-03T13:21:02Z No. of bitstreams: 1 preprint DMUC stucture OPUC.pdf: 208172 bytes, checksum: 70cf9113a3a6852e232eb2e80ab4a285 (MD5) Approved for entry into archive by Pessoa (pfep@ubi.pt) on 2020-02-04T15:54:55Z (GMT) No. of bitstreams: 1 preprint DMUC stucture OPUC.pdf: 208172 bytes, checksum: 70cf9113a3a6852e232eb2e80ab4a285 (MD5) Approved for entry into archive by Pessoa (pfep@ubi.pt) on 2020-02-04T15:57:11Z (GMT) No. of bitstreams: 1 preprint DMUC stucture OPUC.pdf: 208172 bytes, checksum: 70cf9113a3a6852e232eb2e80ab4a285 (MD5) Made available in DSpace on 2020-02-04T15:57:12Z (GMT). No. of bitstreams: 1 preprint DMUC stucture OPUC.pdf: 208172 bytes, checksum: 70cf9113a3a6852e232eb2e80ab4a285 (MD5) Previous issue date: 2012 info:eu-repo/semantics/publishedVersion

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