| 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 ≤p < C.2 Weak Sequential Completeness of L1 Appendix D A WEAK FORM OF RUNGE''S THEOREM Bibliography Research Publications Symbols Index |
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