  # Principles of Fourier Analysis

## 2nd Edition

CRC Press

788 pages | 76 B/W Illus.

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### Description

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.

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

### Subject Categories

##### BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied
MAT037000
MATHEMATICS / Functional Analysis