1st Edition

Harmonic Analysis and Applications

By John J. Benedetto Copyright 1996
    368 Pages
    by CRC Press

    356 Pages
    by CRC Press

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    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
    Analytic Properties of Fourier Transforms
    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)
    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
    A. Real Analysis
    B. Functional Analysis
    C. Fourier Analysis Formulas
    D. Contributors to Fourier Analysis


    Benedetto, John J.