傅立叶级数， fourier series, modern introduction, 第一卷，第二版， gtm 64

The principal aim in writing this book has been to provide an introduction, barely more, to some aspects of Fourier series and related topics in which a liberal use is made of modern techniques and which guides the reader toward some of the problems of current interest in harmonic analysis generally. The use of modern concepts and techniques is, in fact, as wide-spread as is deemed to be compatible with the desire that the book shall be useful to senior undergraduates and beginning graduate students, for whom it may perhaps serve as preparation for Rudin＇s Harmonic Analysis on Groups and the promised second volume of Hewitt and Ross＇s Abstract Harmonic Analysis.

Chapter 1 TRIGONOMETRIC SERIES

　AND FOURIER SERIES

　　1.1 The Genesis of Trigonometric Series and Fourier Series

　　1.2 Pointwise Representation of Functions by Trigonometric Series

　　1.3 New Ideas about Representation

　Exercises

Chapter 2 GROUP STRUCTURE

　AND FOURIER SERIES

　　2.1 Periodic Functions

　　2.2 Translates of Functions. Characters and Exponentials. The Invariant Integral

　　2.3 Fourier Coefficients and Their Elementary Properties

　　2.4 The Uniqueness Theorem and the Density of Trigonometric Polynomials

　　2.5 Remarks on the Dual Problems

　　Exercises

Chapter 3 CONVOLUTIONS OF FUNCTIONS

　3.1 Definition and First Properties of Convolution

　3.2 Approximate Identities for Convolution

　3.3 The Group Algebra Concept

　3.4 The Dual Concepts

　Exercises

Chapter 4 HOMOMORPHISMS OF CONVOLUTION

　ALGEBRAS

　　4.1 Complex Homomorphisms and Fourier Coefficients

　　4.2 Homomorphisms of the Group Algebra

　　Exercises

Chapter 5 THE DIRICHLET AND FEJER KERNELS.

　CESARO SUMMABILITY

　　5.1 The Dirichlet and Fejer Kernels

　　5.2 The Localization Principle

　　5.3 Remarks concerning Summability

　Exercises

Chapter 6 CESARO SUMMABILITY OF FOURIER SERIES

　AND ITS CONSEQUENCES

　　6.1 Uniform and Mean Summability

　　6.2 Applications and Corollaries of 6.1.1

　　6.3 More about Pointwise Summability

　　6.4 Pointwise Summability Almost Everywhere

　　6.5 Approximation by Trigonometric Polynomials

　　6.6 General Comments on Summability of Fourier Series

　　6.7 Remarks on the Dual Aspects

　　Exercises

Chapter 7 SOME SPECIAL SERIES AND THEIR

　APPLICATIONS

　　7.1 Some Preliminaries

　　7.2 Pointwise Convergence of the Series C and S

　　7.3 The Series C and S as Fourier Series

　　7.4 Application to A Z

　　7.5 Application to Factorization Problems

　　Exercises

Chapter 8 FOURIER SERIES IN L2

　8.1 A Minimal Property

　8.2 Mean Convergence of Fourier Series in L2. Parseval''s Formula

　8.3 The Riesz-Fischer Theorem

　8.4 Factorization Problems Again

　8.5 More about Mean Moduli of Continuity

　8.6 Concerning Subsequences of SNf

　8.7 A Z Once Again

　Exercises

Chapter 9　POSITIVE DEFINITE FUNCTIONS

　AND BOCHNER''S THEOREM

　　9.1 Mise-en-Scene

　　9.2 Toward the Bochner Theorem

　　9.3 An Alternative Proof of the Parseval Formula

　　9.4 Other Versions of the Bochner Theorem

　　Exercises

Chapter 10 POINTWISE CONVERGENCE

　OF FOURIER SERIES

　　10.1 Functions of Bounded Variation and Jordan''s Test

　　10.2 Remarks on Other Criteria for Convergence; Dini''s Test

　　10.3 The Divergence of Fourier Series

　　10.4 The Order of Magnitude of sNf. Pointwise Convergence Almost Everywhere

　　10.5 More about the Parseval Formula

　　10.6 Functions with Absolutely Convergent Fourier Series

　　Exercises

Appendix A METRIC SPACES AND BAIRE''S THEOREM

　A.1 Some Definitions

　A.2 Baire''s Category Theorem

　A.3 Corollary

　A.4 Lower Semicontinuous Functions

　A.5 A Lemma

Appendix B CONCERNING TOPOLOGICAL LINEAR SPACES

　B.1 Preliminary Definitions

　B.2 Uniform Boundedness Principles

　B.3 Open Mapping and Closed Graph Theorems

　B.4 The Weak Compaeity Principle

　B.5 The Hahn-Banach Theorem

Appendix C THE DUAL OF Lp 1≤ p < ; WEAK SEQUENTIAL COMPLETENESS OF L1

　C.1 The Dual ofLp 1 ≤