Harmonic Analysis and Applications  book cover
1st Edition

Harmonic Analysis and Applications

ISBN 9780849378799
Published July 29, 1996 by CRC Press
368 Pages

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Book Description

Harmonic analysis plays an essential role in understanding a host of engineering, mathematical, and scientific ideas. In Harmonic Analysis and Applications, the analysis and synthesis of functions in terms of harmonics is presented in such a way as to demonstrate the vitality, power, elegance, usefulness, and the intricacy and simplicity of the subject. This book is about classical harmonic analysis - a textbook suitable for students, and an essay and general reference suitable for mathematicians, physicists, and others who use harmonic analysis.

Throughout the book, material is provided for an upper level undergraduate course in harmonic analysis and some of its applications. In addition, the advanced material in Harmonic Analysis and Applications is well-suited for graduate courses. The course is outlined in Prologue I. This course material is excellent, not only for students, but also for scientists, mathematicians, and engineers as a general reference. Chapter 1 covers the Fourier analysis of integrable and square integrable (finite energy) functions on R. Chapter 2 of the text covers distribution theory, emphasizing the theory's useful vantage point for dealing with problems and general concepts from engineering, physics, and mathematics. Chapter 3 deals with Fourier series, including the Fourier analysis of finite and infinite sequences, as well as functions defined on finite intervals.

The mathematical presentation, insightful perspectives, and numerous well-chosen examples and exercises in Harmonic Analysis and Applications make this book well worth having in your collection.

Table of Contents

Prologue I-Course I Prologue II-Fourier Transforms, Fourier Series, and Discrete Fourier Transforms Fourier Transforms Definitions and Formal Calculations Algebraic Properties of Fourier Transforms Examples Analytic Properties of Fourier Transforms Convolution Approximate Identities and Examples Pointwise Inversion of the Fourier Transform Partial Differential Equations Gibbs Phenomenon The L2(R) Theory Exercises Measures and Distribution Theory Approximate Identities Definition of Distributions Differentiation of Distributions The Fourier Transform of Distributions Convolution of Distributions Operational Calculus Measure Theory Definitions from Probability Theory Wiener's Generalized Harmonic Analysis (GHA) exp{it2} Exercises Fourier Series Fourier Series - Definitions and Convergence History of Fourier Series Integration and Differentiation of Fourier Series The L1(T) and L2(T) Theories A(T) and the Wiener Inversion Theorem Maximum Entropy and Spectral Estimation Prediction and Spectral Estimation Discrete Fourier Transform Fast Fourier Transform Periodization and Sampling Exercises Appendices A. Real Analysis B. Functional Analysis C. Fourier Analysis Formulas D. Contributors to Fourier Analysis Notation Bibliography Index

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Benedetto, John J.