Discrete and Continuous Boundary Problems

Name: Discrete and Continuous Boundary Problems
Brand: Academic Press
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ATKINSON(Author)

ATKINSON(Editor)

Academic Press

Published on 14. May 2014

585 pages

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Front Cover
Discrete and Continuous Boundary Problems
Copyright Page
Contents
Preface
Introduction
0.1. Difference and Differential Equations
0.2. The Invariance Property
0.3. The Scalar Case
0.4. All-Pass Transfer Functions
0.5. Inverse Problems
0.6. The General Orthogonal Case
0.7. The Three-Term Recurrence Formula
0.8. The 2-by-2 Symplectic Case
Chapter 1. Boundary Problems for Rational Functions
1.1. Finite Fourier Series
I .2. The Boundary Problem
1.3. Oscillation Properties
1.4. Eigenfunctions and Orthogonality
1.5. The Spectral Function
1.6. The Characteristic Function
1.7. The First Inverse Problem
1.8. The Second Inverse Problem
1.9. Moment Characterization of the Spectral Function
1.10. Solution of a Moment Problem
Chapter 2. The Infinite Discrete Case
2.1. A Limiting Procedure
2.2. Convergence of the Fundamental Solution
2.3. Convergence of the Spectral Function
2.4. Convergence of the Characteristic Function
2.5. Eigenvalues and Orthogonality
2.6. Orthogonality and Expansion Theorem
2.7. A Continuous Spectrum
2.8. Moment and Interpolation Problem
2.9. A Mixed Boundary Problem
2.10. A Mixed Expansion Problem
2.11. Further Boundary Problems
Chapter 3. Discrete Linear Problems
3.1. Problems Linear in the Parameter
3.2. Reduction to Canonical Form
3.3. The Real Axis Case
3.4. The Unit Circle Case
3.5. The Real 2-by-2 Case
3.6. The 2-by-2 Unit Circle Case
3.7. The Boundary Problem on the Real Axis
3.8. The Boundary Problem on the Unit Circle
Chapter 4. Finite Orthogonal Polynomials
4.1. The Recurrence Relation
4.2. Lagrange-Type Identities
4.3. Oscillatory Properties
4.4. Orthogonality
4.5. Spectral and Characteristic Functions
4.6. The First Inverse Spectral Problem
4.7. The Second Inverse Spectral Problem
4.8. Spectral Functions in General
4.9. Some Continuous Spectral Functions
Chapter 5. Orthogonal Polynomials The Infinite Case
5.1. Limiting Boundary Problems
5.2. Spectral Functions
5.3. Orthogonality and Expansion Theorem
5.4. Nesting Circle Analysis
5.5. Limiting Spectral Functions
5.6. Solutions of Summable Square
5.7. Eigenvalues in the Limit-Circle Case
5.8. Limit-Circle, Limit-Point Tests
5.9. Moment Problem
5.10. The Dual Expansion Theorem
Chapter 6. Matrix Methods for Polynomials
6.1. Orthogonal Polynomials as Jacobi Determinants
6.2. Expansion Theorems, Periodic Boundary Conditions
6.3. Another Method for Separation Theorems
6.4. The Green's Function
6.5. A Reactance Theorem
6.6. Polynomials with Matrix Coefficients
6.7. Oscillatory Properties
6.8. Orthogonality
6.9. Polynomials in Several Variables
6.10. The Multi-Parameter Oscillation Theorem
6.11. Multi-Dimensional Orthogonality
Chapter 7. Polynomials Orthogonal on the Unit Circle
7.1. The Recurrence Relation
7.2. The Boundary Problem
7.3. Orthogonality
7.4. The Recurrence Formulas Deduced from the Orthogonality
7.5. Uniqueness of the Spectral Function
7.6. The Characteristic Function
7.7. A Further Orthogonality Result
7.8. Asymptotic Behavior
7.9. Polynomials Orthogonal on a Real Segment
7.10. Continuous and Discrete Analogs
Chapter 8. Sturm-Liouville Theory
8.1. The Differential Equation
8.2. Existence, Uniqueness, and Bounds for Solutions
8.3. The Boundary Problem
8.4. Oscillatory Properties
8.5. An Interpolatory Property
8.6. The Eigenfunction Expansion
8.7. Second-Order Equation with Discontinuities
8.8. The Green's Function
8.9. Convergence of the Eigenfunction Expansion
8.10. Spectral Functions
8.11. Explicit Expansion Theorem
8.12. Expansions over a Half-Axis
8.13. Nesting Circles
Chapter 9. The General First-Order Differential System
9.1. Formalities
9.2. The Boundary Problem
9.3. Eigenfunctions and Orthogonality
9.4. The Inhomogeneous Problem
9.5. The Characteristic Function
9.6. The Eigenfunction Expansion
9.7. Convergence of the Eigenfunction Expansion
9.8. Nesting Circles
9.9. Expansion of the Basic Interval
9.10. Limit-Circle Theory
9.11. Solutions of Integrable Square
9.12. The Limiting Process a - 8, b + 8
Chapter 10. Matrix Oscillation Theory
10.1. Introduction
10.2. The Matrix Sturm-Liouville Equation
10.3. Separation Theorem for Conjugate Points
10.4. Estimates of Oscillation
10.5. Boundary Problems with a Parameter
10.6. A Fourth-Order Scalar Equation
10.7. The First-Order Equation
10.8. Conjugate Point Problems
10.9. First-Order Equation with Parameter
Chapter 11. From Differential to Integral Equations
11.1. The Sturm-Liouville Case
11.2. Uniqueness and Existence of Solutions
11.3. Wronskian Identities
11.4. Variation of Parameters
11.5. Analytic Dependence on a Parameter
11.6. Eigenvalues and Orthogonality
11.7. Remarks on the Expansion Theorem
11.8. The Generalized First-Order Matrix Differential Equation
11.9. A Special Case
11.10. The Boundary Problem
Chapter 12. Asymptotic Theory of Some Integral Equations
12.1. Asymptotically Trigonometric Behavior
12.2. The S-Function
12.3. A Non-Self-Adjoint Problem
12.4. The Sturm-Liouville Problem
12.5. Asymptotic Properties for the Generalization of y" + [k2 + g(x)]y = 0
12.6. Solutions of Integrable Square
12.7. Analytic Aspects of Asymptotic Theory
12.8. Approximations over a Finite Interval
12.9. Approximation to the Eigenfunctions
12.10. Completeness of the Eigenfunctions
Appendix I: Some Compactness Principles for Stieltjes Integrals
I.1. Functions of Bounded Variation
I.2. The Riemann-Stieltjes Integral
I.3. A Convergence Theorem
I.4. The Helly-Bray Theorem
I.5. Infinite Interval and Bounded Integrand
I.6. Infinite Interval with Polynomial Integrand
I.7. A Periodic Case
I.8. The Matrix Extension
I.9. The Multi-Dimensional Case
Appendix II: Functions of Negative Imaginary Type
II.1. Introduction
II.2. The Rational Case
II.3. Separation Property in the Meromorphic Case
Appendix III: Orthogonality of Vectors
III.1. The Finite-Dimensional Case
III.2. The Infinite-Dimensional Case
Appendix IV: Some Stability Results for Linear Systems
IV.1. A Discrete Case
IV.2. The Case of a Differential Equation
IV.3. A Second-Order Differential Equation
IV.4. The Mixed or Continuous-Discrete Case
IV.5. The Extended Gronwall Lemma
Appendix V: Eigenvalues of Varying Matrices
V.1. Variational Expressions for Eigenvalues
V.2. Continuity and Monotonicity of Eigenvalues
V.3. A Further Monotonicity Criterion
V.4. Varying Unitary Matrices
V.5. Continuation of the Eigenvalues
V.6. Monotonicity of the Unit Circle
Appendix VI: Perturbation of Bases in Hilbert Space
VI.1. The Basic Result
VI.2. Continuous Variation of a Basis
VI.3. Another Result
Notation and Terminology
List of Books and Monographs
Notes
Problems
Index
Mathematics in Science and Engineering

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