 1st Edition

# Harmonic Analysis and Applications

By

## John J. Benedetto

ISBN 9780367401276
Published September 27, 2019 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.

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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## Author(s)

### Biography

Benedetto, John J.