scholarly journals On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle

2020 ◽  
pp. 75-84
Author(s):  
Sergey ZAGORODNYUK
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Sobolev Orthogonal Polynomials

Zeros of Sobolev orthogonal polynomials on the unit circle

2012 ◽  
Vol 60 (4) ◽  
pp. 669-681 ◽  
Author(s):  
K. Castillo ◽  
L. E. Garza ◽  
F. Marcellán
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Sobolev Orthogonal Polynomials

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.

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