Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 30 Nov 2022 07:00:31 GMT2022-11-30T07:00:31Z50171Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circlehttp://hdl.handle.net/10316/11261Title: Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle
Authors: Branquinho, A.; Rebocho, M. N.
Abstract: In this paper we characterize sequences of polynomials on the unit
circle, orthogonal with respect to a Hermitian linear functional such that its corresponding
Carath´eodory function satisfies a Riccati differential equation with polynomial
coefficients, in terms of matrix Sylvester differential equations. Furthermore,
under certain conditions, we give a representation of such sequences in terms of
semi-classical orthogonal polynomials on the unit circle. For the particular case of
semi-classical orthogonal polynomials on the unit circle, a characterization in terms
of first order differential systems is established.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112612008-01-01T00:00:00ZDistributional equation for Laguerre-Hahn functionals on the unit circlehttp://hdl.handle.net/10316/11283Title: Distributional equation for Laguerre-Hahn functionals on the unit circle
Authors: Branquinho, A.; Rebocho, M. N.
Abstract: Let u be a hermitian linear functional defined in the linear space of
Laurent polynomials and F its corresponding Carath´eodory function. We establish
the equivalence between a Riccati differential equation with polynomial coefficients
for F, zAF′ = BF2+CF +D and a distributional equation for u, D(Au) = B1u2+
C1u+H1L, where L is the Lebesgue functional, and the polynomials B1,C1,D1 are
defined in terms of the polynomials A,B,C,D
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10316/112832007-01-01T00:00:00ZProblemas de Momentos e Polinómios ortogonaishttp://hdl.handle.net/10316/13623Title: Problemas de Momentos e Polinómios ortogonais
Authors: Rebocho, Maria das Neves Vieiro
Description: Tese de Mestrado em Matemática,
especialidade em Matemática Pura, ramo de Teoria
da Aproximação, apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra
Mon, 01 Jan 2001 00:00:00 GMThttp://hdl.handle.net/10316/136232001-01-01T00:00:00ZOn the semiclassical character of orthogonal polynomials satisfying structure relationshttp://hdl.handle.net/10316/13712Title: On the semiclassical character of orthogonal polynomials satisfying structure relations
Authors: Branquinho, A.; Rebocho, M. N.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/137122010-01-01T00:00:00ZOn differential equations for orthogonal polynomials on the unit circlehttp://hdl.handle.net/10316/11242Title: On differential equations for orthogonal polynomials on the unit circle
Authors: Branquinho, A.; Rebocho, M. N.
Abstract: In this paper we characterize sequences of orthogonal polynomials on
the unit circle whose corresponding Carath´eodory function satisfies a Riccati differential
equation with polynomial coefficients, in terms of second order matrix
differential equations.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112422008-01-01T00:00:00ZOn Laguerre–Hahn affine orthogonal polynomials on the unit circle from matrix Sylvester equationshttp://hdl.handle.net/10316/43967Title: On Laguerre–Hahn affine orthogonal polynomials on the unit circle from matrix Sylvester equations
Authors: Rebocho, Maria das Neves
Abstract: In this paper are derived recurrences for the reflection coefficients of Laguerre–Hahn affine orthogonal polynomials on the unit circle, including a form of the discrete Painlevé equations dP_V. The technique is based on the knowledge of the first-order differential equation for the Carathéodory function, combined with a re-interpretation, in the formalism of matrix Sylvester equations, of compatibility conditions for the differential systems satisfied by the polynomials.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/439672015-01-01T00:00:00ZOn linear spectral transformations and the Laguerre–Hahn classhttp://hdl.handle.net/10316/43961Title: On linear spectral transformations and the Laguerre–Hahn class
Authors: Castillo, Kenier; Rebocho, Maria das Neves
Abstract: We study the Christoffel and Geronimus transformations for Laguerre–Hahn orthogonal polynomials on the real line. It is analysed the modification on the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recurrence relation coefficients.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/439612017-01-01T00:00:00ZCharacterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform latticeshttp://hdl.handle.net/10316/43842Title: Characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices
Authors: Branquinho, Amílcar; Rebocho, Maria das Neves
Abstract: A characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices is stated and proved. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/438422015-01-01T00:00:00ZSylvester equations for Laguerre–Hahn orthogonal polynomials on the real linehttp://hdl.handle.net/10316/43844Title: Sylvester equations for Laguerre–Hahn orthogonal polynomials on the real line
Authors: Branquinho, Amílcar; Paiva, Anabela; Rebocho, Maria das Neves
Abstract: Matrix Sylvester differential equations are introduced in the study of Laguerre–Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre–Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre–Hahn orthogonal polynomials.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/438442013-01-01T00:00:00ZSecond-order differential equations in the Laguerre–Hahn classhttp://hdl.handle.net/10316/43841Title: Second-order differential equations in the Laguerre–Hahn class
Authors: Branquinho, Amílcar; Foulquié Moreno, Ana; Paiva, Anabela; Rebocho, Maria das Neves
Abstract: Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/438412015-01-01T00:00:00ZDeformed Laguerre–Hahn orthogonal polynomials on the real linehttp://hdl.handle.net/10316/43843Title: Deformed Laguerre–Hahn orthogonal polynomials on the real line
Authors: Branquinho, Amílcar; Rebocho, Maria das Neves
Abstract: One derives discrete dynamical systems related to Laguerre–Hahn orthogonal polynomials. One studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t-dependence on the coefficients of the Riccati differential equation for the related Stieltjes function.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/438432014-01-01T00:00:00ZOrthogonal polynomials on systems of non-uniform lattices from compatibility conditionshttp://hdl.handle.net/10316/43964Title: Orthogonal polynomials on systems of non-uniform lattices from compatibility conditions
Authors: Filipuk, Galina; Rebocho, Maria das Neves
Abstract: We deduce difference equations in the matrix form for Laguerre–Hahn orthogonal polynomials on systems of non-uniform lattices, the so-called compatibility conditions, involving the transfer matrices. As a consequence, we obtain closed form expressions for the recurrence relation coefficients of the Laguerre–Hahn polynomials of class zero.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/439642017-01-01T00:00:00ZDiscrete Painlevé equations for recurrence coefficients of Laguerre–Hahn orthogonal polynomials of class onehttp://hdl.handle.net/10316/43966Title: Discrete Painlevé equations for recurrence coefficients of Laguerre–Hahn orthogonal polynomials of class one
Authors: Filipuk, Galina; Rebocho, Maria das Neves
Abstract: In this paper we study recurrences for Laguerre–Hahn orthogonal polynomials of class one. It is shown for some families of such Laguerre–Hahn polynomials that the coefficients of the three term recurrence relation satisfy some forms of discrete Painlevé equations, namely dP_I and dP_IV.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/439662016-01-01T00:00:00ZDifferential equations for families of semi-classical orthogonal polynomials within class onehttp://hdl.handle.net/10316/44393Title: Differential equations for families of semi-classical orthogonal polynomials within class one
Authors: Filipuk, Galina; Rebocho, Maria das Neves
Abstract: In this paper we study families of semi-classical orthogonal polynomials within class one. We derive general second or third order ordinary differential equations (with respect to certain parameters) for the recurrence coefficients of the three-term recurrence relation of these polynomials and show that in particular well-known cases, e.g. related to the modified Airy and Laguerre weights, these equations can be reduced to the second and the fourth Painlevé equations.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10316/443932018-01-01T00:00:00ZPolinómios ortogonais do tipo Laguerre-Hahn sobre a circunferência unitáriahttp://hdl.handle.net/10316/7515Title: Polinómios ortogonais do tipo Laguerre-Hahn sobre a circunferência unitária
Authors: Rebocho, Maria das Neves Vieiro
Abstract: Neste trabalho temos como objectivo dar um contributo à análise de propriedades diferenciais de famílias de polinómios ortogonais sobre a circunferência unitária. Em particular, centramos o nosso estudo nas famílias de polinómios ortogonais sobre a circunferência unitária e respectivas funcionais de ortogonalidade cujas funções de Carathéodory, F, verificam equações diferenciais de Riccati com coficientes polinomiais, zAF’= BF2+CF+D: Designemos o conjunto das funcionais deste tipo (equivalentemente, o conjunto das sucessões de polinómios ortogonais relativamente a uma funcional deste tipo) de classe Laguerre-Hahn sobre a circunferência unitária. Apresentaremos caracterizações da classe Laguerre-Hahn sobre a circunferência unitária em termos de: - uma equação distribucional para a funcional de ortogonalidade (cf. cap. II); - relações de estrutura de primeira ordem de coeficientes polinomiais (cf. cap. III); - equações diferenciais vectoriais de segunda ordem (cf. cap. III); - equações diferenciais matriciais de Sylvester (cf. cap. IV). Além disso, obteremos uma representação para sucessões de polinómios Laguerre-Hahn sobre a circunferência unitária em termos de famílias semi-clássicas sobre a circunferência unitária (cf. cap. IV).; In this work we aim at giving a contribution to the analysis of differential properties of families of orthogonal polynomials on the unit circle. We focus our study on the families of orthogonal polynomials on the unit circle and corresponding functionals whose Carathéodory functions, F, satisfy Riccati differential equations with polynomial coefficients, zAF’= BF2+CF+D: We shall call the set of such functionals (equivalently, the sequences of polynomials orthogonal with respect to functionals of this kind) the Laguerre-Hahn class on the unit circle. We will give characterizations of the Laguerre-Hahn class on the unit circle in terms of: - a distributional equation to the functional of orthogonality (cf. chapter II); first order structure relations with polynomial cofficients (cf. chapter III); - second order dfferential equations (cf. chapter III); - matrix Sylvester differential equations (cf. chapter IV). Moreover, we will obtain a representation of sequences of Laguerre-Hahn polynomials on the unit circle in terms of semi-classical families on the unit circle (cf. chapter IV).
Description: Tese de doutoramento em Matemática (Matemática Pura) apresentada à Faculdade de Ciências da Universidade de Coimbra
Wed, 16 Jul 2008 00:00:00 GMThttp://hdl.handle.net/10316/75152008-07-16T00:00:00ZCoherent pairs of linear functionals on the unit circlehttp://hdl.handle.net/10316/4581Title: Coherent pairs of linear functionals on the unit circle
Authors: Branquinho, A.; Moreno, A. Foulquié; Marcellán, F.; Rebocho, M. N.
Abstract: In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if ([mu]0,[mu]1) is a coherent pair of measures on the unit circle, then [mu]0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with [mu]1 is a specific rational transformation of the linear functional corresponding to [mu]0. Some examples are given.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45812008-01-01T00:00:00ZCharacterizations of Laguerre-Hahn affine orthogonal polynomials on the unit circlehttp://hdl.handle.net/10316/11309Title: Characterizations of Laguerre-Hahn affine orthogonal polynomials on the unit circle
Authors: Branquinho, A.; Rebocho, M. N.
Abstract: In this work we characterize a monic polynomial sequence, orthogonal
with respect to a hermitian linear functional u that satisfies a functional equation
D(Au) = Bu + zHL, where A,B and H are polynomials and L is the Lebesgue
functional, in terms of a first order linear differential equation for the Carath´eodory
function associated with u and in terms of a first order structure relation for the
orthogonal polynomials.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10316/113092007-01-01T00:00:00Z