A Cohen type inequality for Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer-Sobolev inner product

2011 ◽  
Vol 284 (2-3) ◽  
pp. 240-254 ◽  
Author(s):  
Bujar Xh. Fejzullahu
Keyword(s):  
Orthogonal Polynomials ◽  
Type Inequality ◽  
Inner Product ◽  
Fourier Expansions ◽  
Sobolev Inner Product

scholarly journals Bivariate Koornwinder–Sobolev Orthogonal Polynomials

2021 ◽  
Vol 18 (6) ◽  
Author(s):  
Misael E. Marriaga ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Inner Product ◽  
Recursive Methods ◽  
Sobolev Orthogonal Polynomials ◽  
Sobolev Inner Product

AbstractThe so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function $$\rho (t)$$ ρ ( t ) such that $$\rho (t)^2$$ ρ ( t ) 2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.

scholarly journals On orthogonal polynomials with respect to certain discrete Sobolev inner product

2012 ◽  
Vol 257 (1) ◽  
pp. 167-188 ◽  
Author(s):  
Francisco Marcellán ◽  
Ramadan Zejnullahu ◽  
Bujar Fejzullahu ◽  
Edmundo Huertas
Keyword(s):  
Orthogonal Polynomials ◽  
Inner Product ◽  
Sobolev Inner Product

Orthogonal Polynomials Associated with a Δ-Sobolev Inner Product

2002 ◽  
Vol 8 (2) ◽  
pp. 125-151 ◽  
Author(s):  
María Álvarez De Morales ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Inner Product ◽  
Sobolev Inner Product

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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