scholarly journals On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

2019 ◽  
Vol 53 (2) ◽  
pp. 139-164
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán ◽  
Alejandro Molano
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
Fourier Coefficients ◽  
Infinite Subset ◽  
The Real ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pair ◽  
Coherent Pairs

In the pioneering paper [13], the concept of Coherent Pair was introduced by Iserles et al. In particular, an algorithm to compute Fourier Coefficients in expansions of Sobolev orthogonal polynomials defined from coherent pairs of measures supported on an infinite subset of the real line is described. In this paper we extend such an algorithm in the framework of the so called Symmetric (1, 1)-Coherent Pairs presented in [8].

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

scholarly journals Sylvester equations for Laguerre–Hahn orthogonal polynomials on the real line

2013 ◽  
Vol 219 (17) ◽  
pp. 9118-9131 ◽  
Author(s):  
A. Branquinho ◽  
A. Paiva ◽  
M.N. Rebocho
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
The Real

scholarly journals LOWERING AND RAISING OPERATORS FOR THE FREE MEIXNER CLASS OF ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (03) ◽  
pp. 387-399 ◽  
Author(s):  
EUGENE LYTVYNOV ◽  
IRINA RODIONOVA
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
Meixner Polynomials ◽  
The Real ◽  
Raising Operators ◽  
Meixner Class

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

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
Sign in / Sign up

Export Citation Format

Share Document