From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals

2020 ◽  
pp. 245-292
Author(s):  
Juan C. García-Ardila ◽  
Francisco Marcellán ◽  
Misael E. Marriaga
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Functionals ◽  
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.

scholarly journals Asymptotics of Sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support

1997 ◽  
Vol 81 (2) ◽  
pp. 217-227 ◽  
Author(s):  
Francisco Marcellán ◽  
Andrei Martínez-Finkelshtein ◽  
Juan J. Moreno-Balcázar
Keyword(s):  
Orthogonal Polynomials ◽  
Compact Support ◽  
Sobolev Orthogonal Polynomials ◽  
Coherent Pairs

scholarly journals Spectral Transformations and Associated Linear Functionals of the First Kind

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 107
Author(s):  
Juan Carlos García-Ardila ◽  
Francisco Marcellán
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Space ◽  
Linear Functional ◽  
Linear Functionals ◽  
Spectral Transformations ◽  
Christoffel Transformation ◽  
Complex Coefficients

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials

2003 ◽  
Vol 34 (2-4) ◽  
pp. 203-216 ◽  
Author(s):  
A.C. Berti ◽  
C.F. Bracciali ◽  
A. Sri Ranga
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Orthogonal Polynomials

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

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

scholarly journals General Sobolev Orthogonal Polynomials

1996 ◽  
Vol 200 (3) ◽  
pp. 614-634 ◽  
Author(s):  
Francisco Marcellán ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar ◽  
André Ronveaux
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Orthogonal Polynomials
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