Principles of Fourier Analysis
Features
Summary
Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas.
Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented.
Principles of Fourier Analysis stimulates the appreciation and understanding of the fundamental concepts and serves both beginning students who have seen little or no Fourier analysis as well as the more advanced students who need a deeper understanding. Insightful, non-rigorous derivations motivate much of the material, and thought-provoking examples illustrate what can go wrong when formulas are misused. With clear, engaging exposition, readers develop the ability to intelligently handle the more sophisticated mathematics that Fourier analysis ultimately requires.
Table of Contents
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
