scholarly journals Sobolev Orthogonal Polynomials on the Unit Circle and Coherent Pairs of Measures of the Second Kind

2016 ◽  
Vol 71 (3-4) ◽  
pp. 1127-1149
Author(s):  
F. Marcellán ◽  
A. Sri Ranga
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pairs

scholarly journals Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

2014 ◽  
Vol 96 (110) ◽  
pp. 193-210 ◽  
Author(s):  
Francisco Marcellán ◽  
Natalia Pinzón-Cortés
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Linear Functional ◽  
Inner Product ◽  
Complex Numbers ◽  
Linear Functionals ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pair ◽  
Coherent Pairs ◽  
Sobolev Inner Product

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 {?n(z)}n>0 and {?n(z)}n?0 satisfy ?Mi=0 ai,n?(m) n+m?i(z) = ?Nj=0 bj,n?n?j(z), n ? 0, where M,N,m ? 0, ai,n and bj,n, for 0 ? i ? M, 0 ? j ? N, n > 0, are complex numbers such that aM,n ? 0, n ? M, bN,n ? 0, n ? N, and ai,n = bi,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)?= (U,p(z)q(1/z))+ ?(V, p(m)(z)q(m)(1/z)), ? > 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-Szeg? linear functional. Finally, a matrix interpretation of (M,N)-coherence is given.

scholarly journals Asymptotics for Jacobi–Sobolev orthogonal polynomials associated with non-coherent pairs of measures

2010 ◽  
Vol 162 (11) ◽  
pp. 1945-1963 ◽  
Author(s):  
Eliana X.L. de Andrade ◽  
Cleonice F. Bracciali ◽  
Laura Castaño-García ◽  
Juan J. Moreno-Balcázar
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pairs

Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs

2018 ◽  
Vol 467 (1) ◽  
pp. 601-621 ◽  
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán Español ◽  
Alejandro Molano Molano
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pairs

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Garza ◽  
Francisco Marcellán ◽  
Natalia C. Pinzón-Cortés
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Linear Functional ◽  
Linear Functionals ◽  
Spectral Transformation ◽  
Coherent Pair ◽  
Coherent Pairs

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.

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

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 246 ◽  
Author(s):  
Lino G. Garza ◽  
Luis E. Garza ◽  
Edmundo J. Huertas
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Recurrence Relation ◽  
Second Order ◽  
Connection Formula ◽  
Second Order Differential Equations ◽  
Spectral Transformations ◽  
Structure Formula ◽  
Coherent Pairs

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.

Sign in / Sign up

Export Citation Format

Share Document