Fourier Series: A Modern Introduction Volume 1 - R. E. Edwards

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Contentss.- 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.- 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.- 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.- 4 Homomorphisms of Convolution Algebras.- 4.1 Complex Homomorphisms and Fourier Coefficients.- 4.2 Homomorphisms of the Group Algebra.- Exercises.- 5 The Dirichlet and Fej?r Kernels. Ces?ro Summability.- 5.1 The Dirichlet and Fej?r Kernels.- 5.2 The Localization Principle.- 5.3 Remarks concerning Summability.- Exercises.- 6 Ces?ro Summability of Fourier Series and its Consequences.- 6.1 Uniform and Mean Summability.- 6.2 Applications and Corollaries of.1.1 90.- 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.- 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.- 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.- 9 Positive Definite Functions and Bochner's Theorem.- 9.1 Mise-en-Sc?ne.- 9.2 Toward the Bochner Theorem.- 9.3 An Alternative Proof of the Parseval Formula.- 9.4 Other Versions of the Bochner Theorem.- Exercises.- 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 Compacity Principle.- B.5 The Hahn-Banach Theorem.- Appendix D A WEAK FORM OF RUNGE'S THEOREM.- Research Publications.- Symbols.