Principles of fourier analysis

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Principles of fourier analysis / Kenneth B. Howell

資料種別:: 図書
出版情報:: Boca Raton, FL : Chapman & Hall/CRC, c2001
形態:: 776 p. ; 26 cm
シリーズ名:: Studies in advanced mathematics <BA1341974X>
著者名:: Howell, Kenneth B.
ISBN:: 9780849382758 [0849382750]
書誌ID:: BA52587245

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注記:

Includes bibliographical references (p. 757) and index

主題:

Fourier analysis

言語:

英語

目次情報:

Preface

Sample Courses

Preliminaries / I：

The Starting Point / 1：

Fourier's Bold Conjecture / 1.1：

Mathematical Preliminaries and the Following Chapters / 1.2：

Basic Terminology, Notation, and Conventions / 2：

Numbers / 2.1：

Functions, Formulas, and Variables / 2.2：

Operators and Transforms / 2.3：

Basic Analysis I: Continuity and Smoothness / 3：

(Dis)Continuity / 3.1：

Differentiation / 3.2：

Basic Manipulations and Smoothness / 3.3：

Addenda / 3.4：

Basic Analysis II: Integration and Infinite Series / 4：

Integration / 4.1：

Infinite Series (Summations) / 4.2：

Symmetry and Periodicity / 5：

Even and Odd Functions / 5.1：

Periodic Functions / 5.2：

Sines and Cosines / 5.3：

Elementary Complex Analysis / 6：

Complex Numbers / 6.1：

Complex-Valued Functions / 6.2：

The Complex Exponential / 6.3：

Functions of a Complex Variable / 6.4：

Functions of Several Variables / 7：

Basic Extensions / 7.1：

Single Integrals of Functions with Two Variables / 7.2：

Double Integrals / 7.3：

Addendum / 7.4：

Fourier Series / II：

Heuristic Derivation of the Fourier Series Formulas / 8：

The Frequencies / 8.1：

The Coefficients / 8.2：

Summary / 8.3：

The Trigonometric Fourier Series / 9：

Defining the Trigonometric Fourier Series / 9.1：

Computing the Fourier Coefficients / 9.2：

Partial Sums and Graphing / 9.3：

Fourier Series over Finite Intervals (Sine and Cosine Series) / 10：

The Basic Fourier Series / 10.1：

The Fourier Sine Series / 10.2：

The Fourier Cosine Series / 10.3：

Using These Series / 10.4：

Inner Products, Norms, and Orthogonality / 11：

Inner Products / 11.1：

The Norm of a Function / 11.2：

Orthogonal Sets of Functions / 11.3：

Orthogonal Function Expansions / 11.4：

The Schwarz Inequality for Inner Products / 11.5：

Bessel's Inequality / 11.6：

The Complex Exponential Fourier Series / 12：

Derivation / 12.1：

Notation and Terminology / 12.2：

Computing the Coefficients / 12.3：

Partial Sums / 12.4：

Convergence and Fourier's Conjecture / 13：

Pointwise Convergence / 13.1：

Uniform and Nonuniform Approximations / 13.2：

Convergence in Norm / 13.3：

The Sine and Cosine Series / 13.4：

Convergence and Fourier's Conjecture: The Proofs / 14：

Basic Theorem on Pointwise Convergence / 14.1：

Convergence for a Particular Saw Function / 14.2：

Convergence for Arbitrary Saw Functions / 14.3：

Derivatives and Integrals of Fourier Series / 15：

Differentiation of Fourier Series / 15.1：

Differentiability and Convergence / 15.2：

Integrating Periodic Functions and Fourier Series / 15.3：

Sine and Cosine Series / 15.4：

Applications / 16：

The Heat Flow Problem / 16.1：

The Vibrating String Problem / 16.2：

Functions Defined by Infinite Series / 16.3：

Verifying the Heat Flow Problem Solution / 16.4：

Classical Fourier Transforms / III：

Heuristic Derivation of the Classical Fourier Transform / 17：

Riemann Sums over the Entire Real Line / 17.1：

The Derivation / 17.2：

Integrals on Infinite Intervals / 17.3：

Absolutely Integrable Functions / 18.1：

The Set of Absolutely Integrable Functions / 18.2：

Many Useful Facts / 18.3：

Functions with Two Variables / 18.4：

The Fourier Integral Transforms / 19：

Definitions, Notation, and Terminology / 19.1：

Near-Equivalence / 19.2：

Linearity / 19.3：

Invertibility / 19.4：

Other Integral Formulas (A Warning) / 19.5：

Some Properties of the Transformed Functions / 19.6：

Classical Fourier Transforms and Classically Transformable Functions / 20：

The First Extension / 20.1：

The Set of Classically Transformable Functions / 20.2：

The Complete Classical Fourier Transforms / 20.3：

What Is and Is Not Classically Transformable? / 20.4：

Duration, Bandwidth, and Two Important Sets of Classically Transformable Functions / 20.5：

More on Terminology, Notation, and Conventions / 20.6：

Some Elementary Identities: Translation, Scaling, and Conjugation / 21：

Translation / 21.1：

Scaling / 21.2：

Practical Transform Computing / 21.3：

Complex Conjugation and Related Symmetries / 21.4：

Differentiation and Fourier Transforms / 22：

The Differentiation Identities / 22.1：

Rigorous Derivation of the Differential Identities / 22.2：

Higher Order Differential Identities / 22.3：

Anti-Differentiation and Integral Identities / 22.4：

Gaussians and Other Very Rapidly Decreasing Functions / 23：

Basic Gaussians / 23.1：

General Gaussians / 23.2：

Gaussian-Like Functions / 23.3：

Complex Translation and Very Rapidly Decreasing Functions / 23.4：

Convolution and Transforms of Products / 24：

Derivation of the Convolution Formula / 24.1：

Basic Formulas and Properties of Convolution / 24.2：

Algebraic Properties / 24.3：

Computing Convolutions / 24.4：

Existence, Smoothness, and Derivatives of Convolutions / 24.5：

Convolution and Fourier Analysis / 24.6：

Correlation, Square-Integrable Functions, and the Fundamental Identity of Fourier Analysis / 25：

Correlation / 25.1：

Square-Integrable/Finite Energy Functions / 25.2：

The Fundamental Identity / 25.3：

Identity Sequences / 26：

An Elementary Identity Sequence / 26.1：

General Identity Sequences / 26.2：

Gaussian Identity Sequences / 26.3：

Verifying Identity Sequences / 26.4：

An Application (with Exercises) / 26.5：

Generalizing the Classical Theory: A Naive Approach / 27：

Delta Functions / 27.1：

Transforms of Periodic Functions / 27.2：

Arrays of Delta Functions / 27.3：

The Generalized Derivative / 27.4：

Fourier Analysis in the Analysis of Systems / 28：

Linear, Shift-Invariant Systems / 28.1：

Computing Outputs for LSI Systems / 28.2：

Gaussians as Test Functions, and Proofs of Some Important Theorems / 29：

Testing for Equality with Gaussians / 29.1：

The Fundamental Theorem on Invertibility / 29.2：

The Fourier Differential Identities / 29.3：

The Fundamental and Convolution Identities of Fourier Analysis / 29.4：

Generalized Functions and Fourier Transforms / IV：

A Starting Point for the Generalized Theory / 30：

Starting Points / 30.1：

Gaussian Test Functions / 31：

The Space of Gaussian Test Functions / 31.1：

On Using the Space of Gaussian Test Functions / 31.2：

Two Other Test Function Spaces and a Confession / 31.3：

More on Gaussian Test Functions / 31.4：

Norms and Operational Continuity / 31.5：

Generalized Functions / 32：

Functionals / 32.1：

Basic Algebra of Generalized Functions / 32.2：

Generalized Functions Based on Other Test Function Spaces / 32.4：

Some Consequences of Functional Continuity / 32.5：

The Details of Functional Continuity / 32.6：

Sequences and Series of Generalized Functions / 33：

Sequences and Limits / 33.1：

A Little More on Delta Functions / 33.2：

Basic Transforms of Generalized Fourier Analysis / 33.4：

Fourier Transforms / 34.1：

Generalized Scaling of the Variable / 34.2：

Generalized Translation/Shifting / 34.3：

Transforms of Limits and Series / 34.4：

Adjoint-Defined Transforms in General / 34.6：

Generalized Complex Conjugation / 34.7：

Generalized Products, Convolutions, and Definite Integrals / 35：

Multiplication and Convolution / 35.1：

Definite Integrals of Generalized Functions / 35.2：

Appendix: On Defining Generalized Products and Convolutions / 35.3：

Periodic Functions and Regular Arrays / 36：

Periodic Generalized Functions / 36.1：

Fourier Series for Periodic Generalized Functions / 36.2：

On Proving Theorem 36.5 / 36.3：

General Solutions to Simple Equations and the Pole Functions / 37：

Basics on Solving Simple Algebraic Equations / 37.1：

Homogeneous Equations with Polynomial Factors / 37.2：

Nonhomogeneous Equations with Polynomial Factors / 37.3：

The Pole Functions / 37.4：

Pole Functions in Transforms, Products, and Solutions / 37.5：

The Discrete Theory / V：

Periodic, Regular Arrays / 38：

The Index Period and Other Basic Notions / 38.1：

Fourier Series and Transforms of Periodic, Regular Arrays / 38.2：

Sampling and the Discrete Fourier Transform / 39：

Some General Conventions and Terminology / 39.1：

Sampling and the Discrete Approximation / 39.2：

The Discrete Approximation and Its Transforms / 39.3：

The Discrete Fourier Transforms / 39.4：

Discrete Transform Identities / 39.5：

Fast Fourier Transforms / 39.6：

Appendices

Fourier Transforms of Some Common Functions / Table A.1：

Identities for the Fourier Transforms / Table A.2：

References

Answers to Selected Exercises

Index

Preface

Sample Courses

Preliminaries / I：

The Starting Point / 1：

Fourier's Bold Conjecture / 1.1：

Mathematical Preliminaries and the Following Chapters / 1.2：

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