The principal aim in writing this book has been to provide an intro­ duction, barely more, to some aspects of Fourier series and related topics in which a liberal use is made of modem techniques and which guides the reader toward some of the problems of current interest in harmonic analysis generally. The use of modem 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. The emphasis on modem techniques and outlook has affected not only the type of arguments favored, but also to a considerable extent the choice of material. Above all, it has led to a minimal treatment of pointwise con­ vergence and summability: as is argued in Chapter 1, Fourier series are not necessarily seen in their best or most natural role through pointwise-tinted spectacles. Moreover, the famous treatises by Zygmund and by Baryon trigonometric series cover these aspects in great detail, wl:tile leaving some gaps in the presentation of the modern viewpoint; the same is true of the more elementary account given by Tolstov. Likewise, and again for reasons discussed in Chapter 1, trigonometric series in general form no part of the program attempted.

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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 Fejér Kernels. Cesàro Summability.- 5.1 The Dirichlet and Fejér Kernels.- 5.2 The Localization Principle.- 5.3 Remarks concerning Summability.- Exercises.- 6 Cesàro 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-Scène.- 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.