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.

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 Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

2019 ◽  
Vol 373 (2) ◽  
pp. 875-917 ◽  
Author(s):  
Alexander I. Aptekarev ◽  
Sergey A. Denisov ◽  
Maxim L. Yattselev
Keyword(s):  
Orthogonal Polynomials ◽  
Jacobi Matrices ◽  
Multiple Orthogonal Polynomials

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles

2009 ◽  
Vol 29 (6) ◽  
pp. 1881-1905 ◽  
Author(s):  
S. JITOMIRSKAYA ◽  
D. A. KOSLOVER ◽  
M. S. SCHULTEIS
Keyword(s):  
Lyapunov Exponent ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Quantum Dynamics ◽  
Jacobi Matrices

AbstractIt is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

scholarly journals A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 213
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán ◽  
Alejandro Molano
Keyword(s):  
Positive Definite ◽  
Linear Functionals ◽  
Definite Case ◽  
Coherent Pairs

In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described.

Sign in / Sign up

Export Citation Format

Share Document