Orthogonal Polynomials and Linear Functionals

2021 ◽  
Author(s):  
Juan Carlos García-Ardila ◽  
Francisco Marcellán ◽  
Misael E. Marriaga
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Functionals

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.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
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.

scholarly journals Orthogonal polynomials for refinable linear functionals

2006 ◽  
Vol 75 (256) ◽  
pp. 1891-1903 ◽  
Author(s):  
Dirk Laurie ◽  
Johan de Villiers
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Functionals

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

2021 ◽  
Author(s):  
Juan 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 the 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 the Geronimus transformations.

scholarly journals Complex Jacobi Matrices and Gauss Quadrature for Quasi-definite Linear Functionals

2015 ◽  
Author(s):  
Stefano Pozza ◽  
Miroslav Pranić ◽  
Zdenek Strakoš
Keyword(s):  
Orthogonal Polynomials ◽  
Positive Definite ◽  
Gauss Quadrature ◽  
Jacobi Matrices ◽  
Linear Functionals ◽  
Underlying Theory ◽  
Close Relationship

The Gauss quadrature can be formulated as a method for approximation of positive definite linear functionals. The underlying theory connects several classical topics including orthogonal polynomials and (real) Jacobi matrices. In the poster we investigated the problem of generalizing the concept of Gauss quadrature for approximation of linear functionals which are not positive definite. We showed that the concept can be generalized to quasi-definite functionals and based on a close relationship with orthogonal polynomials and complex Jacobi matrices.

Matrix factorizations and orthogonal polynomials

2019 ◽  
Vol 09 (01) ◽  
pp. 2040003 ◽  
Author(s):  
Diego Dominici
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Factorizations ◽  
Linear Functionals ◽  
Meixner Polynomials ◽  
Matrix Decompositions

We present some elements of the theory of orthogonal polynomials based on matrix decompositions. We focus our attention on discrete linear functionals, and use the Meixner polynomials as a concrete example.

scholarly journals Laguerre–Hahn orthogonal polynomials on the real line

2019 ◽  
Vol 09 (01) ◽  
pp. 2040001
Author(s):  
Maria Das Neves Rebocho
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Real Line ◽  
Linear Functionals ◽  
Recurrence Coefficients ◽  
Stieltjes Functions ◽  
The Real ◽  
Polynomial Coefficients ◽  
Type Differential Equation

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

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