Orthogonal Polynomials: Current Trends and Applications

Name: Orthogonal Polynomials: Current Trends and Applications | Proceedings of the 7th EIBPOA Conference
Brand: Springer
Price: 149.79 EUR
Availability: OnlineOnly

Proceedings of the 7th EIBPOA Conference

Francisco Marcellán Edmundo J. Huertas(Herausgeber*in)

Springer (Verlag)

Erschienen am 11. Februar 2021

VIII, 327 Seiten

E-Book

PDF mit Wasserzeichen-DRM

Systemvoraussetzungen

978-3-030-56190-1 (ISBN)

149,79 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Wasserzeichen-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Inhalt

Intro
Preface
Contents
Riemann-Hilbert Problem and Matrix Biorthogonal Polynomials
1 Introduction
2 Riemann-Hilbert Problem for Matrix Biorthogonal Polynomials
2.1 Matrix Biorthogonal Polynomials
2.2 Reductions: From Biorthogonality to Orthogonality
2.3 The Riemann-Hilbert Problem
2.4 Three Term Recurrence Relation
3 Matrix Weights Supported on a Curve ? on the Complex Plane that Connects the Point 0 to the Point 8: Laguerre Weights
3.1 Matrix Weights Supported on a Curve ? with One Finite End Point: W(z) = zA H(z)
3.2 Matrix Weights Supported on a Curve ? with One Finite End Point: W(z) = za H(z)G(z) zB
References
The Symmetrization Problem for Multiple Orthogonal Polynomials
1 Introduction
2 Definitions and Matrix Interpretation of Multiple Orthogonality
3 Representation of the Stieltjes and Weyl Functions
4 The Symmetrization Problem for Multiple OP
5 Matrix Multiple Orthogonality
6 Example with Multiple Hermite Polynomials
6.1 The Matrix Type II Multiple Orthogonal Polynomials
6.2 The Symmetric Type II Multiple Orthogonal Polynomial Sequence
References
Darboux Transformations for Orthogonal Polynomials on the Real Line and on the Unit Circle
1 Introduction
2 Orthogonal Polynomials and Matrix Representations
2.1 OPRL and Jacobi Matrices
2.2 OPUC and Hessenberg Matrices
2.3 OLPUC and CMV Matrices
3 The Darboux Transformations
3.1 Darboux Transformations of Self-Adjoint Operators
3.2 Darboux Transformations of Unitary Operators
4 Darboux Transformations of Jacobi Matrices
4.1 Inverse Darboux Transformations of Jacobi Matrices
5 Darboux Transformations of CMV Matrices
5.1 Inverse Darboux Transformations of CMV Matrices
References
Special Function Solutions of Painlevé Equations: Theory, Asymptotics and Applications
1 Painlevé Equations
1.1 Hamiltonian Formulation and Tau Functions
1.2 Bäcklund Transformations and Toda Equation
1.3 Special Function Solutions of Painlevé Equations
2 Motivation 1. Orthogonal Polynomials
2.1 General Theory
2.2 Connection with Painlevé Equations
2.3 The Riemann-Hilbert Formulation
3 Motivation 2: Random Matrix Theory
4 Asymptotic Analysis
4.1 Asymptotics as n8
4.2 Asymptotics as z8
5 Conclusions and Related Problems
References
Discrete Semiclassical Orthogonal Polynomials of Class 2
1 Introduction
2 Basic Background on Discrete Semiclassical Linear Functionals
3 Modification of Functionals
3.1 Rational Spectral Transformations
3.1.1 Uvarov Transformations
3.1.2 Christoffel Transformations
3.1.3 Geronimus Transformations
3.2 Truncated Linear Functionals
3.3 Symmetrized Functionals
3.4 Summary
4 Semiclassical Polynomials of Class 0 (Classical Polynomials)
4.1 (0,0): Charlier Polynomials
4.2 (1,0): Meixner Polynomials
4.3 (1,0
N): Krawtchouk Polynomials
4.3.1 Symmetrized Charlier Polynomials
4.4 (2,1
N,1): Hahn Polynomials
4.4.1 Symmetrized Meixner Polynomials
4.4.2 Symmetrized Generalized Charlier Polynomials
5 Semiclassical Polynomials of Class 1
5.1 (0,1): Generalized Charlier Polynomials
5.2 (1,1): Generalized Meixner Polynomials
5.2.1 Reduced-Uvarov Charlier Polynomials
5.2.2 Christoffel Charlier Polynomials
5.2.3 Geronimus Charlier Polynomials
5.2.4 Truncated Charlier Polynomials
5.3 (2,0
N): Generalized Krawtchouk Polynomials
5.4 (2,1): Generalized Hahn Polynomials of Type I
5.4.1 Reduced-Uvarov Meixner Polynomials
5.4.2 Christoffel Meixner Polynomials
5.4.3 Geronimus Meixner Polynomials
5.4.4 Truncated Meixner Polynomials
5.4.5 Symmetrized Generalized Krawtchouk Polynomials
5.5 (3,2
N,1): Generalized Hahn Polynomials of Type II
5.5.1 Reduced-Uvarov Hahn Polynomials
5.5.2 Christoffel Hahn Polynomials
5.5.3 Geronimus Hahn Polynomials
5.5.4 Symmetrized Hahn Polynomials
5.5.5 Symmetrized Polynomials of Type (3,0
N)
6 Semiclassical Polynomials of Class 2
6.1 Polynomials of Type (0,2)
6.2 Polynomials of Type (1,2)
6.2.1 Reduced-Uvarov Generalized Charlier Polynomials
6.2.2 Christoffel Generalized Charlier Polynomials
6.2.3 Geronimus Generalized Charlier Polynomials
6.2.4 Truncated Generalized Charlier Polynomials
6.3 Polynomials of Type (2,2)
6.3.1 Uvarov Charlier Polynomials
6.3.2 Reduced-Uvarov Generalized Meixner Polynomials
6.3.3 Christoffel Generalized Meixner Polynomials
6.3.4 Geronimus Generalized Meixner Polynomials
6.3.5 Truncated Generalized Meixner Polynomials
6.4 Polynomials of Type (3,0
N)
6.5 Polynomials of Type (3,1
N)
6.5.1 Reduced-Uvarov Generalized Krawtchouk Polynomials
6.5.2 Christoffel Generalized Krawtchouk Polynomials
6.5.3 Geronimus Generalized Krawtchouk Polynomials
6.6 Polynomials of Type (3,2)
6.6.1 Uvarov Meixner Polynomials
6.6.2 Symmetrized Generalized Meixner Polynomials
6.6.3 Reduced-Uvarov Generalized Hahn Polynomials of Type I
6.6.4 Christoffel Generalized Hahn Polynomials of Type I
6.6.5 Geronimus Generalized Hahn Polynomials of Type I
6.6.6 Truncated Generalized Hahn Polynomials of Type I
6.6.7 Symmetrized Polynomials of Type (0,2)
6.7 Polynomials of Type (4,3
N,1)
6.7.1 Uvarov Hahn Polynomials
6.7.2 Symmetrized Generalized Hahn Polynomials of Type I
6.7.3 Reduced-Uvarov Generalized Hahn Polynomials of Type II
6.7.4 Christoffel Generalized Hahn Polynomials of Type II
6.7.5 Geronimus Generalized Hahn Polynomials of Type II
6.7.6 Symmetrized Polynomials of Type (1,2)
7 Conclusions
References
Stable Equilibria for the Roots of the Symmetric Continuous Hahn and Wilson Polynomials
1 Introduction
2 Symmetric Continuous Hahn Polynomials
2.1 Preliminaries
2.2 Morse Function
2.3 Gradient Flow
2.4 Proof of Theorem 1
2.5 Numerical Samples
3 Wilson Polynomials
3.1 Preliminaries
3.2 Morse Function
3.3 Gradient Flow
3.4 Proof of Theorem 2
3.5 Numerical Samples
4 Symmetry Reduction
References
Convergent Non Complete Interpolatory Quadrature Rules
1 Introduction
2 Some Explicit Convergent Schemes of Nodes
3 Connection with Orthogonal Polynomials
4 Asymptotic Analysis
5 Proof of Theorem 1
References
Asymptotic Computation of Classical Orthogonal Polynomials
1 Introduction
2 Laguerre Polynomials
3 Jacobi Polynomials
3.1 An Expansion in Terms of Elementary Functions
3.2 An Expansion in Terms of Bessel Functions
3.3 An Expansion for Large ß
3.4 An Expansion for Large a and ß
3.5 Computational Performance
Appendix
References
An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation
1 Introduction
1.1 Some Historical Background
1.2 Markov Systems and Orthogonality
1.2.1 Angelesco Systems
1.2.2 Nikishin Systems
2 On the Perfectness of Nikishin Systems
3 On the Interlacing Property of Zeros
3.1 Interlacing for Type I
3.2 Interlacing for Type II
4 Weak Asymptotic
4.1 Preliminaries from Potential Theory
4.2 Weak Asymptotic Behavior for Type II
4.3 Application to Hermite-Padé Approximation
5 Ratio Asymptotic
5.1 Preliminaries from Riemann Surfaces and Boundary Value Problems
References
Revisiting Biorthogonal Polynomials: An LU Factorization Discussion
1 Introduction
2 LU Factorization and Gram Matrix
2.1 LDU Factorization
2.2 Schur Complements
2.3 Bilinear Forms, Gram Matrices and LU Factorizations
3 Orthogonal Polynomials
3.1 Quasi-Determinants
3.1.1 The Easiest Quasi-Determinant: A Schur Complement
3.2 Second Kind Functions and LU Factorizations
3.3 Spectral Matrices
3.4 Christoffel-Darboux Kernels
4 Standard Orthogonality: Hankel Reduction
4.1 Recursion Relations
4.2 Heine Formulas
4.3 Gauss Quadrature Formula
4.4 Christoffel-Darboux Formula
5 Very Classical Orthogonal Polynomials: Hermite, Laguerre and Jacobi Polynomials
6 Christoffel and Geronimus Transformations
6.1 Some History
6.2 Christoffel and Geronimus Transformations
6.3 Christoffel Perturbations
6.4 Geronimus Transformations
References
Infinite Matrices in the Theory of Orthogonal Polynomials
1 Introduction
2 The Algebra of Infinite Lower Hessenberg Matrices
3 Polynomial Sequences and Their Matrix Representation
4 Orthogonal Polynomial Sequences
5 Orthogonal Polynomial Sequences That Satisfy a Generalized First-Order Difference Equation
6 Hypergeometric Orthogonal Polynomial Sequences
7 A Family of q-Orthogonal Polynomial Sequences That Contains All the Families in the q-Askey Scheme
8 Final Remarks
References

Systemvoraussetzungen

Als PDF speichern Als Link merken