Principles of Fourier Analysis

PRELIMINARIES 
The Starting Point 
Basic Terminology, Notation, and Conventions 
Basic Analysis I: Continuity and Smoothness 
Basic Analysis II: Integration and Infinite Series 
Symmetry and Periodicity 
Elementary Complex Analysis 
Functions of Several Variables 
FOURIER SERIES 
Heuristic Derivation of the Fourier Series Formulas 
The Trigonometric Fourier Series 
Fourier Series over Finite Intervals (Sine and Cosine Series) 
Inner Products, Norms, and Orthogonality 
The Complex Exponential Fourier Series 
Convergence and Fourier`s Conjecture 
Convergence and Fourier`s Conjecture: The Proofs 
Derivatives and Integrals of Fourier Series 
Applications 
CLASSICAL FOURIER TRANSFORMS 
Heuristic Derivation of the Classical Fourier Transform 
Integrals on Infinite Intervals 
The Fourier Integral Transforms 
Classical Fourier Transforms and Classically Transformable Functions 
Some Elementary Identities: Translation, Scaling, and Conjugation 
Differentiation and Fourier Transforms 
Gaussians and Other Very Rapidly Decreasing Functions
Convolution and Transforms of Products 
Correlation, Square-Integrable Functions, and the Fundamental Identity of Fourier Analysis 
Identity Sequences 
Generalizing the Classical Theory: A Naive Approach 
Fourier Analysis in the Analysis of Systems 
Gaussians as Test Functions, and Proofs of Some Important Theorems 
GENERALIZED FUNCTIONS AND FOURIER TRANSFORMS 
A Starting Point for the Generalized Theory 
Gaussian Test Functions 
Generalized Functions 
Sequences and Series of Generalized Functions 
Basic Transforms of Generalized Fourier Analysis 
Generalized Products, Convolutions, and Definite Integrals 
Periodic Functions and Regular Arrays 
General Solutions to Simple Equations and the Pole Functions
THE DISCRETE THEORY 
Periodic, Regular Arrays 
Sampling and the Discrete Fourier Transform 
APPENDICES