Coherent pairs of linear functionals on the unit circle

Branquinho, A.; Foulquié-Moreno, Ana; Marcellán, Francisco; Rebocho, M. N.

doi:10.1016/j.jat.2008.03.003

Cited by 14 publications

(10 citation statements)

References 5 publications

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“…In Section 3, we will introduce the Sobolev inner product (1.2) p(z), q(z) λ = U, p(z) q(1/z) +λ V, p (m) (z) q (m) (1/z) , λ > 0, m ∈ Z + , and we will study its corresponding sequence of monic orthogonal polynomials when the regular Hermitian linear functionals U and V form an (M, N )-coherent pair of order m. In this way, we will highlight the cases (M, N ) = (1, 1) and (M, N ) = (1, 0). These results generalize those given by Branquinho, Foulquié-Moreno, Marcellán, and Rebocho in [2]. Finally, as an example, we will consider the cases when U is the Lebesgue linear functional and V is the Bernstein-Szegő linear functional.…”

Section: Introductionsupporting

confidence: 70%

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

Marcellán

Pinzón-Cortés

2014

Publ Inst Math (Belgr)

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A pair of regular Hermitian linear functionals (U , V) is said to be an (M, N)-coherent pair of order m on the unit circle if their corresponding sequences of monic orthogonal polynomials {φn(z)} n 0 and {ψn(z)} n 0 satisfy M i=0 a i,n φ (m) n+m−i (z) = N j=0 b j,n ψ n−j (z), n 0, where M, N, m 0, a i,n and b j,n , for 0 i M , 0 j N , n 0, are complex numbers such that a M,n = 0, n M , b N,n = 0, n N , and a i,n = b i,n = 0, i > n. When m = 1, (U , V) is called a (M, N)-coherent pair on the unit circle. We focus our attention on the Sobolev inner product p(z), q(z) λ = U , p(z)q(1/z) + λ V, p (m) (z)q (m) (1/z) , λ > 0, m ∈ Z + , assuming that U and V is an (M, N)-coherent pair of order m on the unit circle. We generalize and extend several recent results of the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. Besides, we analyze the cases (M, N) = (1, 1) and (M, N) = (1, 0) in detail. In particular, we illustrate the situation when U is the Lebesgue linear functional and V is the Bernstein-Szegő linear functional. Finally, a matrix interpretation of (M, N)-coherence is given.

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Section: Introductionsupporting

confidence: 70%

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

Marcellán

Pinzón-Cortés

2014

Publ Inst Math (Belgr)

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“…Finally, Alvarez-Nodarse et al [13] analyzed the more general case, ( , )--coherent pairs of order ( , ) and ( , )-coherent pairs of order ( , ), proving the analogue results to those in [4]. Furthermore, Branquinho et al in [14] extended the concept of coherent pair to Hermitian linear functionals associated with nontrivial probability measures supported on the unit circle. They studied (3) in the framework of orthogonal polynomials on the unit circle (OPUC).…”

Section: Introductionmentioning

confidence: 91%

(1,1)-Coherent Pairs on the Unit Circle

Garza

Marcellán

Pinzón-Cortés

2013

Abstract and Applied Analysis

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A pair(𝒰,𝒱)of Hermitian regular linear functionals on the unit circle is said to be a(1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials{ϕn(x)}n≥0and{ψn(x)}n≥0satisfyϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1, whereϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when𝒰is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where𝒱is associated with either a positive nontrivial measure or its rational spectral transformation.

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“…then (µ 0 , µ 1 ) is said to be a (1,0)-coherent pair on the unit circle. It was shown in [30] that (36) constitutes a sufficient condition for…”

Section: An Uvarov Perturbation Of the Circular Jacobi Polynomialsmentioning

confidence: 99%

“…The notion of coherent pairs of orthogonality measures on the unit circle was introduced (in the more general framework of linear functionals defined in the space of Laurent polynomials) in [30], where the authors considered the Sobolev inner product…”

Section: A Christoffel Transformation Of the Bernstein-szegő Polynomialsmentioning

confidence: 99%

On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle

Garza

Huertas

2020

Mathematics

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In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle. most 0, 1, and 2, respectively. He also found that the Hermite, Jacobi, and Laguerre families are the only polynomial solutions for (1) with a corresponding measure that is positive. Moreover, such sequences satisfy the so-called Hahn's property: their derivatives also constitute an orthogonal family. It is interesting that some of these families have been studied in the context of symmetry Lie algebras. For instance, in [3][4][5][6], the authors studied how the differential equations satisfied by the classical families can be obtained from the second-order Casimir elements of the corresponding symmetry algebra and of some of its subalgebras. Furthermore, some operators defined on Lie algebras are used in [7] to obtain differential properties of some special functions, including the Jacobi polynomials.On the other hand, in the last decades, some canonical examples of spectral transformations of orthogonality measures have been studied in the literature: the Christoffel transformation, consisting in a polynomial modification of the measure; the Uvarov transformation, defined by the addition of a Dirac's delta measure; and the Geronimus transformation, where the orthogonality measure is divided by a polynomial, and a Dirac's delta measure is added. For instance, in [8], the author is interested in the relations between the associated Stieltjes functions. Furthermore, he shows that all linear spectral transformations of Stieltjes functions can be obtained as a finite product of the three canonical transformations mentioned above. On the other hand, in [9], the authors consider the relations between these perturbations and LU factorizations of the corresponding Jacobi matrices, in the more general framework of orthogonality with respect to linear functionals. The study of differential equations of higher order satisfied by orthogonal polynomials was initiated by H. L. Krall in [10] and later on A. M. Krall (see [11]) showed that the sequences orthogonal with respect to an Uvarov perturbation of the classical families satisfy a fourth order differential equation. The orthogonal families associated with these perturbations are often called classical-type orthogonal polynomials. It is important to notice that they are not longer classical, and the polynomial coefficients in the corresponding differential equations (which are called holonomic equations) can even depend on the degree n (see [12]), in sharp contrast with the classical case...

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Coherent pairs of linear functionals on the unit circle

Cited by 14 publications

References 5 publications

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

(1,1)-Coherent Pairs on the Unit Circle

On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle

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