傅立叶级数 第1卷 第2版（影印版）

　　 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

　　 PREFACE
　　 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.
　　 The emphasis on modern 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 convergence 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 Bary on trigonometric series cover these aspects in great detail, while leaving some gaps in the presentation of the modern viewpoint; the same is true of the more elementary account given by Tolstoy. Likewise, and again for reasons discussed in Chapter 1, trigonometric series in general form no part of the program attempted.
　　 A considerable amount of space has been devoted to matters that cannot in a book of this size and scope receive detailed treatment. Among such material, much of which appears in small print, appear comments on diverse specialized topics (capacity, spectral synthesis sets, Helson sets, and so forth), as well as remarks on extensions of results to more general groups.The object in including such material is, in the first case, to say enough for the reader to gain some idea of the meaning and significance of the problems involved, and to provide a guide to further reading; and in the second case,to provide some sort of "cultural" background stressing a unity that
　　underlies apparently diverse fields. It cannot be over-emphasized that the book is perforce introductory in all such matters.
　　 The demands made in terms of the reader's active cooperation increase fairly steadily with the chapter numbers, and although the book is surely best regarded as a whole, Volume I is self-contained, is easier than Volume II,and might be used as the basis of a short course. In such a short course, it would be feasible to omit Chapter 9 and Section 10.6.
　　 As to specific requirements made of the reader, the primary and essential item is a fair degree of familiarity with Lebesgue integration to at least the extent described in Williamson's introductory book Lebesgue Integration. Occasionally somewhat more is needed, in which case reference is made to Appendix C, to Hewitt and Stromberg's Real and Abstract Analysis, or to
　　Asplund and Bungart's A First Course in Integration. In addition, the reader needs to know what metric spaces and normed linear spaces are, and to have some knowledge of the rudiments of point-set topology. The remaining results in functional analysis (category arguments, uniform boundedness principles, the closed graph, open mapping, and Hahn-Banach theorems)are dealt with in Appendixes A and B. The basic terminology of linear algebra is used, but no result of any depth is assumed.
　　 Exercises appear at the end of each chapter, the more difficult ones being provided with hints to their solutions.
　　 The bibliography, which refers to both book and periodical literature,contains many suggestions for further reading in almost all relevant directions and also a sample of relevant research papers that have appeared since the publication of the works by Zygmund, Bary, and Rudin already cited. Occasionally, the text contains references to reviews of periodical literature.
　　 My first acknowledgment is to thank Professors Hanna Neumann and Edwin Hewitt for encouragement to begin the book, the former also for the opportunity to try out early drafts of Volume I on undergraduate students in the School of General Studies of the Australian National University, and the latter also for continued encouragement and advice. My thanks are due
　　also to the aforesaid students for corrections to the early drafts.
　　 In respect to the technical side of composition, I am extremely grateful to my colleague, Dr. Garth Gaudry, who read the entire typescript (apart from last-minute changes) with meticulous care, made innumerable valuable suggestions and vital corrections, and frequently dragged me from the brink of disaster. Beside this, the compilation of Sections 13.7 and 13.8 and Subsection 13.9.1 is due entirely to him. Since, however, we did not always agree on minor points of presentation, I alone must take the blame for shortcomings of this nature. To him I extend my warmest thanks.
　　 My thanks are offered to Mrs. Avis Debnam, Mrs. K. Sumeghy, and Mrs.Gail Liddell for their joint labors on the typescript.
　　 Finally, I am deeply in debt to my wife for all her help with the proofreading and her unfailing encouragement.
　　 R.E.E.
　　 CANBERRA, 1967