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Preface
Contents
Index
LINEAR MATHEMATICS
IN
INFINITE DIMENSIONS
Signals
Boundary Value Problems
and Special Functions
U. H. Gerlach
Date:
Febuary 2007
Beta Edition
Preface
Contents
Introduction
Infinite Dimensional Vector Spaces
Inner Product Spaces
Normed Linear Spaces
Metric Spaces
Complete Metric Spaces
Limit of a Sequence
Cauchy Sequence
Cauchy Completeness: Complete Metric Space, Banach Space, and Hilbert Space
Hilbert Spaces
Two Prototypical Examples
The Riesz-Fischer Theorem
Orthogonal Basis and Orthogonalization
Approximation via Subspaces: Analysis and Geometry
Subspace Approximation vs. the Riesz-Fischer Theorem
Recapitulation
Isomorphic Hilbert Spaces
Fourier Theory
The Dirichlet Kernel
Basic Properties
Three Applications
Solution to Wave Equation via Dirichlet Kernel
Dirichlet Kernel: Fountainhead of All Subspace Vectors
Whittaker-Shannon Sampling Theorem: The Finite Interval Version
Fourier Series of a Function
The Dirac Delta Function
The Fourier Integral
Transition from Fourier Series to Fourier Integral
The Fourier Integral Theorem
The Fourier Transform as a Unitary Transformation
Fourier Transform via Parseval's Relation
Efficient Calculation: Fourier Transform via Convolution
Orthonormal Wave Packet Representation
Orthonormal Wave Packets: General Construction
Orthonormal Wave Packets: Definition and Properties
Definition
Four Properties
Whittaker-Shannon Sampling Theorem: The Infinite Interval Version
Phase Space Representation
Orthonormal Wavelet Representation
Construction and Properties
Multiresolution Analysis
Chirped Signals and the Principle of Unit-Economy
Irregular Signals and Variable Resolution Analysis
Multiresolution Analysis as Hierarchical
Central Approximation Space
Translation Followed by Compression
Resolution Spaces as Hierarchical
Resolution Analysis as a Decomposition into Subspaces
Unit-Economy via the Two Parent Wavelets
Multiscale Analysis as a Method of Measurement
Multiscale Analysis vs Multiresolution Analysis: MSA or MRA?
The Pyramid Algorithm
The Requirement of Commensurability
The Scaling Function as a MSA Identifier
Wavelet Analysis
Sturm-Liouville Theory
Three Archetypical Linear Problems
The Homogeneous Problem
Sturm-Liouville Systems
Sturm-Liouville Differential Equation
Homogeneous Boundary Conditions
Basic Properties of a Sturm-Liouville Eigenvalue Problem
Orthogonality, Reality, and Uniqueness
Sturm's Comparison Theorem
Phase Analysis of a Linear Second Order O.D.E.
The Prüfer System
Qualitative Results
Phase Analysis of a Sturm-Liouville System
The Boundary Conditions
The Boundary Value Problem
The Behavior of the Phase: The Oscillation Theorem
Discrete Unbounded Sequence of Eigenvalues
Completeness of the Set of Eigenfunctions via Rayleigh's Quotient
Green's Function Theory
The Adjoint of an Operator
Adjoint Boundary Conditions
Second Order Operator and the Bilinear Concomitant
Green's Function and Its Adjoint
Translation Invariant Systems
Pictorial Definition of a Green's Function
The Simple String and Poisson's Equation
Point Force Applied to the System
Properties and Utility of a Green's Function
Construction of the Green's Function
Unit Impulse Response: General Homogeneous Boundary Conditions
The Totally Inhomogeneous Boundary Value Problem
Spectral Representation
Spectral Resolution of the Resolvent of a Matrix
Spectral Resolution of the Green's Function
Green's Function as the Fountainhead of the Eigenvalues and Eigenvectors of a System
Spectral Representation
Completeness via Green's function
String with Free Ends: Green's Function, Spectrum, and Completeness
Green's Function
Spectrum via Green's Function
Completeness
Boundary Value Problem
via Green's Function: Integral Equation
One-dimensional Scattering Problem: Exterior Boundary Value Problem
One-dimensional Cavity Problem: Interior Boundary Value Problem
Eigenfunctions via Integral Equations
Types of Integral Equations
Singular Boundary Value Problem:
Infinite Domain
Review: Branches, Branch Cuts, and Riemann Sheets
Square Integrability
Infinite String
Infinite String as the Limit of a Finite String
Spectral Representation of the Dirac Delta Function
Coalescence of Poles into a Branch Cut
Contour Integration Around the Branch Cut
Fourier Sine Theorem
Special Function Theory
The Helmholtz Equation
Cartesian versus Polar Coordinates
Degenerate Eigenvalues
Complete Set of Commuting Operators
Translations and Rotations in the Euclidean Plane
Symmetries of the Helmholtz Equations
Wanted: Rotation Invariant Solutions to the Helmholtz Equation
Properties of Hankel and Bessel Functions
Applications of Hankel and Bessel Functions
Exterior Boundary Value Problem: Scattering
Finite Interior Boundary Value Problem: Cavity Vibrations
Infinite Interior Boundary Value Problem: Waves Propagating in a Cylindrical Pipe
More Properties of Hankel and Bessel Functions
The Method of Steepest Descent and Stationary Phase
Boundary Value Problems in Two Dimensions
Solution via Green's Function
Green's Function via Dimensional Reduction
Green's Function: 2-D Laplace vs. (Limit of) 2-D Helmholtz
Wave Equation for Spherically Symmetric Systems
Spherically Symmetric Solutions
Factorization Method for Solving a Partial Differential Equation: Spherical Harmonics
Static Solutions
Static Multipole Field
Additional Theorem for Spherical Harmonics
The Bessel-Legendre Connection
Partial Differential Equations
Single Partial Differential Equations: Their Origin
Boundary Conditions of a Typical Partial Differential Equation in Two Dimensions
Cauchy Problem and Characteristics
Hyperbolic Equations
Riemann's Method for Integrating the Most General 2nd Order Linear Hyperbolic Equation
1. Identify Cauchy Data in Characteristic Form
2. Determine Green's Identity for the Given Differential Operator
3. Apply Green's Identity
System of Partial Differential Equations: How to Solve Maxwell's Equations Using Linear Algebra
Maxwell Wave Equation
The Overdetermined System
Maxwell Wave Equation (continued)
2+2 Decomposition
Divergenceless Vector Fields
The Eigenvector Fields of
The Electric and the Magnetic Fields
Cylindrical Coordinates
The TE Field
The TM Field
The TEM Field
Spherical Coordinates
Left Null Space
Eigenvector Fields
Electromagnetic Fields in a Spherical Coordinate System
Index
About this document ...
Ulrich Gerlach 2007-04-05
