傅立叶级数 第2卷 第2版（影印版）

　　 Apart from a number of minor corrections and changes, a substantial reformulation and up-dating of Chapters 14 and 15 has taken place. This reformulation and up-dating is a major and very welcome contribution from my friend and colleague, Dr J.W. Sanders, to whom I
　　express my sincere thanks. His efforts have produced a much better result than I could have achieved on my own. Warm thanks are also due to Dr Jo Ward, Who checked some of the revised material.
　　 New Sections 16.9 and 16.10 have also been added.
　　 The bibliography has been expanded and brought up to date, though it is still not exhaustive.
　　 In spite of these changes, the third paragraph in the Preface to the revised edition of Volume I is applicable here. What has been aeeomplished here is not a complete account of developments over the past 15 years; such an account would require many volumes. Even so, it
　　may assist some readers who wish to appraise some of these developments. More ambitious readers should consult Mathematical Reviews from around Volume 50 onwards.
　　

chapter 11 spans of translates. closed ideals.
closed subalgebras. banach algebras
11.1 closed invariant subspaces and closed ideals
11.2 the structure of closed ideals and related topics
11.3 closed subalgebras
11.4 banach algebras and their applications
exercises
chapter 12 distributions and measures
12.1 concerning cd
12.2 definition and examples of distributions and measures
12.3 convergence of distributions
12.4 differentiation of distributions
12.5 fourier coefficients and fourier series of distributions
12.6 convolutions of distributions
12.7 more about m and lp
12.8 hilbert's distribution and conjugate series
12.9 the theorem of marcel riesz
12.10 mean convergence of fourier series in lp (1 [ p [ oo)
12.11 pseudomeasures and their applications
12.12 capacities and beurling's problem

　　PREFACE TO THE SECOND (REVISED)
　　 EDITION OF VOLUME 2
　　 Apart from a number of minor corrections and changes, a substantial reformulation and up-dating of Chapters 14 and 15 has taken place. This reformulation and up-dating is a major and very welcome contribution from my friend and colleague, Dr J.W. Sanders, to whom I
　　express my sincere thanks. His efforts have produced a much better result than I could have achieved on my own. Warm thanks are also due to Dr Jo Ward, Who checked some of the revised material.
　　 New Sections 16.9 and 16.10 have also been added.
　　 The bibliography has been expanded and brought up to date, though it is still not exhaustive.
　　 In spite of these changes, the third paragraph in the Preface to the revised edition of Volume I is applicable here. What has been aeeomplished here is not a complete account of developments over the past 15 years; such an account would require many volumes. Even so, it
　　may assist some readers who wish to appraise some of these developments. More ambitious readers should consult Mathematical Reviews from around Volume 50 onwards.
　　 R.E.E.
　　CANBERRA, September 1981
　　 PREFACE TO VOLUME 2
　　 The substance of the first three paragraphs of the preface to Volume i of Fourier Series: A Modern Introduction applies equally well to this second volume. To what is said there, the following remarks should be added.
　　 Volume 2 deals on the whole with the more modem aspects of Fourier theory, and with those facets of the classical theory that fit most naturally into a function-analytic garb. With their introduction to distributional concepts and techniques and to interpolation theorems, respec-
　　tively, Chapters 12 and 13 are perhaps the most significant portions of Volume 2. From a pedagogical viewpoint, the carefully detailed dis*cussion of Marcinkiewicz's interpolation theorem will, it is hoped, go some way toward making this topic more accessible to a beginner.
　　 A major portion of Chapter ll is devoted to the elements of Banach algebra theory and its applications in harmonic analysis. In Chapter 16 there appears what is believed to be the first reasonably connected introductory account of multiplier problems and related matters.
　　 For the purposes of a short course, one might be content to cover Section 11.1, the beginning of Section 11.2, Section 11.4, Chapter 12 up to and including Section 12.10, Chapter 13 up to and including Section 13.6,Chapter 14, and Sections 15.1 to 15.3. Much of Chapters 13 to 15 is
　　independent of Chapters ll and 12, or is easily made so. While severe pruning might lead to a tolerable excision of Section 11.4, which is required but rarely in subsequent chapters, it would be a pity thus to omit all reference to Banach algebras.
　　 I at one time cherished the hope of including in this volume a list of current research problems, but the available space will not accommodate such a list together with the necessary explanatory notes. The interested reader may go a long way toward repairing this defect by studying some of the articles appearing in [Bi] (see, most especially, pp. 351-354 thereof).
　　 The cross-referencing system is as follows. With the exception of references to the appendixes, the numerical component of every reference to either volume appears in the form a ~ b - c, where a, b, and c are positive integers; the material referred to appears in Volume 1 if and only if
　　1 ≤ a ≤ 10. In the case of references to the appendixes, all of which appear in Volume 1, a Roman numeral "I" has been prefixed as a reminder to the reader; thus, for example, "I,B.2.1" refers to Appendix B.2.1 in Volume 1.