傅立葉級數(shù)（第1卷）

內(nèi)容概要

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.

書籍目錄

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　ExercisesChapter 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　　ExercisesChapter 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　ExercisesChapter 4 HOMOMORPHISMS OF CONVOLUTION　ALGEBRAS　　4.1 Complex Homomorphisms and Fourier Coefficients　　4.2 Homomorphisms of the Group Algebra　　ExercisesChapter 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　ExercisesChapter 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　　ExercisesChapter 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　　ExercisesChapter 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　ExercisesChapter 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　　ExercisesChapter 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　　ExercisesAppendix 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 LemmaAppendix 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 TheoremAppendix C THE DUAL OF Lp 1≤ p < ; WEAK SEQUENTIAL COMPLETENESS OF L1　C.1 The Dual ofLp 1 ≤p

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