Principles of Fourier Analysis

ISBN-13: 9781498734097 / Angielski / Twarda / 2016 / 804 str.

Kenneth B. Howell

Opis

Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. This book provides students and practitioners with a practical and mathematically solid introduction to its principles. Principles of Fourier Analysis provides a comprehensive overview of the mathematical theory of Fourier analysis.

Spis treści

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 FunctionsConvolution 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 FunctionsTHE DISCRETE THEORY Periodic, Regular Arrays Sampling and the Discrete Fourier Transform APPENDICES

O autorze

Kenneth Howell is an Associate Professor Emeritus in the Department of Mathematical Sciences of the University of Alabama in Huntsville. He holds a Ph.D. from Indiana University and earned bachelor degrees in both mathematics and physics. Dr. Howell has done extensive work in both academia and in industry. He is also the author of Ordinary Differential Equation: An Introduction to the Fundamentals, also by Chapman & Hall/CRC Press.