Δ-Coherent pairs of linear functionals and Markov-Bernstein inequalities

In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described.

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

Abstract and Applied Analysis ◽

10.1155/2013/307974 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 2

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.

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Coherent pairs of linear functionals on the unit circle

Journal of Approximation Theory ◽

10.1016/j.jat.2008.03.003 ◽

2008 ◽

Vol 153 (1) ◽

pp. 122-137 ◽

Cited By ~ 7

Author(s):

A. Branquinho ◽

A. Foulquié Moreno ◽

F. Marcellán ◽

M.N. Rebocho

Keyword(s):

Unit Circle ◽

Linear Functionals ◽

Coherent Pairs

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(M,N)-Coherent pairs of linear functionals and Jacobi matrices

Applied Mathematics and Computation ◽

10.1016/j.amc.2014.01.055 ◽

2014 ◽

Vol 232 ◽

pp. 76-83 ◽

Cited By ~ 1

Author(s):

Francisco Marcellán ◽

Natalia Camila Pinzón-Cortés

Keyword(s):

Jacobi Matrices ◽

Linear Functionals ◽

Coherent Pairs

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Coherent pairs of measures and Markov–Bernstein inequalities

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2017.01.055 ◽

2017 ◽

Vol 450 (2) ◽

pp. 996-1028 ◽

Cited By ~ 1

Author(s):

André Draux

Keyword(s):

Bernstein Inequalities ◽

Coherent Pairs

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Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

Publications de l Institut Mathematique ◽

10.2298/pim1410193m ◽

2014 ◽

Vol 96 (110) ◽

pp. 193-210 ◽

Cited By ~ 1

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.

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