基本信息
- 原书名：An Introduction to Harmonic Analysis,Third Edition
- 原出版社： Cambridge University Press
- 作者： （美）Yitzhak Katznelson
- 丛书名： 经典原版书库
- 出版社：机械工业出版社
- ISBN：9787111158295
- 上架时间：2005-1-31
- 出版日期：2005 年1月
- 开本：16开
- 页码：314
- 版次：3-1
- 所属分类：数学 > 分析 > 傅里叶分析与小波分析
教材
内容简介
书籍
数学书籍
“本书在具体材料与抽象概念之间取得了恰到好处的平衡，作者在选择最适宜的主题方面独具慧眼。对题材清晰而简练的阐述以及选用的大量练习，对于想要掌握这门学科基础的人来说无疑是理想的知识源泉。”
——2002 Steele Prize Citation
“总而言之，作者所撰写的是一部处于支配地位的、内容丰富的和令读者为之兴奋的著作。无疑它将激励大量对调和分析的进一步研究，并将在这一主题的文献领域中占据重要的位置。”
———Math Revoews
本书自1968年出版后，就牢固地树立了其经典地位，并受到学生和专家们的推崇。Katznelson教授因此书而获得了2002年度的斯提尔奖。
本书从经典傅里叶分析的清晰表述入手，旨在用一个具体的构架展示调和分析的中心思想，并提供了大量有助于透彻了解调和分析理论的例子。作者在确立这些思想之后，转向扩展调和分析，使之远远超越圆群的范围，并通过在实线上讨论傅里叶变换以及在局部紧阿贝尔群上对傅里叶分析的简单考察，打开通向其他领域的大门。
与以前的版本相比，本版增加了若干补充材料，其中包括逼近论中的诸多主题和在调和分析中运用概率论方法的一些例子。
数学书籍
“本书在具体材料与抽象概念之间取得了恰到好处的平衡，作者在选择最适宜的主题方面独具慧眼。对题材清晰而简练的阐述以及选用的大量练习，对于想要掌握这门学科基础的人来说无疑是理想的知识源泉。”
——2002 Steele Prize Citation
“总而言之，作者所撰写的是一部处于支配地位的、内容丰富的和令读者为之兴奋的著作。无疑它将激励大量对调和分析的进一步研究，并将在这一主题的文献领域中占据重要的位置。”
———Math Revoews
本书自1968年出版后，就牢固地树立了其经典地位，并受到学生和专家们的推崇。Katznelson教授因此书而获得了2002年度的斯提尔奖。
本书从经典傅里叶分析的清晰表述入手，旨在用一个具体的构架展示调和分析的中心思想，并提供了大量有助于透彻了解调和分析理论的例子。作者在确立这些思想之后，转向扩展调和分析，使之远远超越圆群的范围，并通过在实线上讨论傅里叶变换以及在局部紧阿贝尔群上对傅里叶分析的简单考察，打开通向其他领域的大门。
与以前的版本相比，本版增加了若干补充材料，其中包括逼近论中的诸多主题和在调和分析中运用概率论方法的一些例子。
作译者
Yitzhak Katznelson于巴黎大学获得博士学位。他曾执教于加州大学伯克利分校、希伯来大学和耶鲁大学，现任斯坦福大学数学教授。他的数学研究领域包括调和分析、遍历理论和可微分动力系统。
目录
I Fourier Series on T
1 Fourier coefficients
2 Summability in norm and homogeneous banach spaces on T
3 Pointwise convergence of n(f)
4 The order of magnitude of Fourier coefficients
5 Fourier series of square summable functions
6 Absolutely convergent Fourier series
7 Fourier coefficients of linear functionals
8 Additional comments and applications
9 The d-dimensional torus
II The Convergence of Fourier Series
1 Convergence in norm
2 Convergence and divergence at a point
3 Sets of divergence
III The Conjugate Function
1 The conjugate function
2 The maximal function of Hardy and Littlewood
3 The Hardy spates
IV Interpolation of Linear Operators
1 Interpolation of norms and of linear operators
1 Fourier coefficients
2 Summability in norm and homogeneous banach spaces on T
3 Pointwise convergence of n(f)
4 The order of magnitude of Fourier coefficients
5 Fourier series of square summable functions
6 Absolutely convergent Fourier series
7 Fourier coefficients of linear functionals
8 Additional comments and applications
9 The d-dimensional torus
II The Convergence of Fourier Series
1 Convergence in norm
2 Convergence and divergence at a point
3 Sets of divergence
III The Conjugate Function
1 The conjugate function
2 The maximal function of Hardy and Littlewood
3 The Hardy spates
IV Interpolation of Linear Operators
1 Interpolation of norms and of linear operators
前言
Harmonic analysis is the study of objects (functions, measures, etc.),defined on topological groups. The group structure enters into the study by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space. The study consists of twosteps. First: finding the "elementary components" of the object, that is, objects of the same or similar class, which exhibit the simplest behavior under translation and which "belong" to the object under study (harmonic or spectral analysis); and second: finding a way in which the object can be construed as a combination of its elementary components (harmonic or spectral synthesis).
The vagueness of this description is due not only to the limitation of the author but also to the vastness of its scope. In trying to make it clearer, one can proceed in various ways*; we have chosen here to sacrifice generality for the sake of concreteness. We start with the circle group T and deal with classical Fourier series in the first five chapters, turning then to the real line in Chapter VI and coming to locally compact abelian groups, only for a brief sketch, in Chapter VII. The philosophy behind the choice of this approach is that it makes it easier for students to grasp the main ideas and gives them a large class of concrete examples which are essential for the proper understanding of the theory in the general context of topological groups. The presentation of Fourier series and integrals differs from that in [1], [7], [8], and [28] in being, I believe, more explicitly aimed at the general (locally compact abelian) case.
The last chapter is an introduction to the theory of commutative Ba-nach algebras. It is biased, studying Banach algebras mainly as a tool in harmonic analysis.
This book is an expanded version of a set of lecture notes written
*Hence the indefinite article in the title of the book.
for a course which I taught at Stanford University during the spring and summer quarters of 1965. The course was intended for graduate students who had already had two quarters of the basic "real-variable" course. The book is on the same level: the reader is assumed to be familiar with the basic notions and facts of Lebesgue integration, the most elementary facts concerning Borel measures, some basic facts about holomorphic functions of one complex variable, and some elements of functional analysis, namely: the notions of a Banach space, continuous linear functionals, and the three key theorems--"the closed graph", the Hahn-Banach, and the "uniform boundedhess" theorems. All the prerequisites can be found in [23] and (except, for the complex variable) in [22]. Assuming these prerequisites, the book, or most of it, can be covered in a one-year course. A slower moving course or one shorter than a year may exclude some of the starred sections (or subsections). Aiming for a one-year course forced the omission not only of the more general setup (non-abelian groups are not even mentioned), but also of many concrete topics such as Fourier analysis on Rn, n > l, and finer problems of harmonic analysis in T or R (some of which can be found in [13]). Also, some important material was cut into exercises, and we urge the reader to do as many of them as he can.
The bibliography consists mainly of books, and it is through the bib-liographies included in these books that the reader is to become familiar with the many research papers written on harmonic analysis. Only some, more recent, papers are included in our bibliography. In general we credit authors only seldom--most often for identification purposes. With the growing mobility of mathematicians, and the happy amount of oral communication, many results develop within the mathematical folklore and when they find their way into print it is not always easy to determine who deserves the credit. When I was writing Chapter III of this book, I was very pleased to produce the simple elegant proof of Theorem III. 1.6 there. I could swear I did it myself until I remembered two days later that six months earlier, "over a cup of coffee," Lennart Carleson indicated to me this same proof.
The book is divided into chapters, sections, and subsections. The chapter numbers are denoted by roman numerals and the sections and subsections, as well as the exercises, by arabic numerals. In cross references within the same chapter, the chapter number is omitted; thus Theorem III. 1.6, which is the theorem in subsection 6 of Section 1 of Chapter III, is referred to as Theorem 1.6 within Chapter III, and Theorem III. 1.6 elsewhere. The exercises are gathered at the end of the sections, and exercise V. 1.1 is the first exercise at the end of Section 1, Chapter V. Again, the chapter number is omitted when an exercise is referred to within the same chapter. The ends of proofs are marked by a triangle ().
The book was written while I was visiting the University of Paris and Stanford University and it owes its existence to the moral and technical help I was so generously given in both places. During the writing I have benefited from the advice and criticism of many friends; I would like to thank them all here. Particular thanks are due to L. Carleson, K. DeLeeuw, J.-P. Kahane, O.C. McGehee, and W. Rudin. I would also like to thank the publisher for the friendly cooperation in the production of this book.
YITZHAK KATZNELSON
Jerusalem
April 1968
The third edition
The second edition was essentially identical with the first, except for the correction of a few misprints. In the current edition some more misprints were corrected, the wording changed in a few places, and some material added: two additional sections in Chapter I and one in Chapter IV; an additional appendix; and a few additional exercises.
The added material does not reflect the progress in the field in the past thirty or forty years. Much of it could and, in retrospect, should have been included in the first edition of the book.
This book was and is intended to serve, as its title makes explicit, as an introduction. It offers what I believe to be the core material and technique, a basis on which much can be built.
The added items in the bibliography expand on parts which are dis-cussed here only briefly (or not at all), and provide a much more up-todate bibliography of Harmonic analysis.
Y.K.
Stanford
June 2003
The vagueness of this description is due not only to the limitation of the author but also to the vastness of its scope. In trying to make it clearer, one can proceed in various ways*; we have chosen here to sacrifice generality for the sake of concreteness. We start with the circle group T and deal with classical Fourier series in the first five chapters, turning then to the real line in Chapter VI and coming to locally compact abelian groups, only for a brief sketch, in Chapter VII. The philosophy behind the choice of this approach is that it makes it easier for students to grasp the main ideas and gives them a large class of concrete examples which are essential for the proper understanding of the theory in the general context of topological groups. The presentation of Fourier series and integrals differs from that in [1], [7], [8], and [28] in being, I believe, more explicitly aimed at the general (locally compact abelian) case.
The last chapter is an introduction to the theory of commutative Ba-nach algebras. It is biased, studying Banach algebras mainly as a tool in harmonic analysis.
This book is an expanded version of a set of lecture notes written
*Hence the indefinite article in the title of the book.
for a course which I taught at Stanford University during the spring and summer quarters of 1965. The course was intended for graduate students who had already had two quarters of the basic "real-variable" course. The book is on the same level: the reader is assumed to be familiar with the basic notions and facts of Lebesgue integration, the most elementary facts concerning Borel measures, some basic facts about holomorphic functions of one complex variable, and some elements of functional analysis, namely: the notions of a Banach space, continuous linear functionals, and the three key theorems--"the closed graph", the Hahn-Banach, and the "uniform boundedhess" theorems. All the prerequisites can be found in [23] and (except, for the complex variable) in [22]. Assuming these prerequisites, the book, or most of it, can be covered in a one-year course. A slower moving course or one shorter than a year may exclude some of the starred sections (or subsections). Aiming for a one-year course forced the omission not only of the more general setup (non-abelian groups are not even mentioned), but also of many concrete topics such as Fourier analysis on Rn, n > l, and finer problems of harmonic analysis in T or R (some of which can be found in [13]). Also, some important material was cut into exercises, and we urge the reader to do as many of them as he can.
The bibliography consists mainly of books, and it is through the bib-liographies included in these books that the reader is to become familiar with the many research papers written on harmonic analysis. Only some, more recent, papers are included in our bibliography. In general we credit authors only seldom--most often for identification purposes. With the growing mobility of mathematicians, and the happy amount of oral communication, many results develop within the mathematical folklore and when they find their way into print it is not always easy to determine who deserves the credit. When I was writing Chapter III of this book, I was very pleased to produce the simple elegant proof of Theorem III. 1.6 there. I could swear I did it myself until I remembered two days later that six months earlier, "over a cup of coffee," Lennart Carleson indicated to me this same proof.
The book is divided into chapters, sections, and subsections. The chapter numbers are denoted by roman numerals and the sections and subsections, as well as the exercises, by arabic numerals. In cross references within the same chapter, the chapter number is omitted; thus Theorem III. 1.6, which is the theorem in subsection 6 of Section 1 of Chapter III, is referred to as Theorem 1.6 within Chapter III, and Theorem III. 1.6 elsewhere. The exercises are gathered at the end of the sections, and exercise V. 1.1 is the first exercise at the end of Section 1, Chapter V. Again, the chapter number is omitted when an exercise is referred to within the same chapter. The ends of proofs are marked by a triangle ().
The book was written while I was visiting the University of Paris and Stanford University and it owes its existence to the moral and technical help I was so generously given in both places. During the writing I have benefited from the advice and criticism of many friends; I would like to thank them all here. Particular thanks are due to L. Carleson, K. DeLeeuw, J.-P. Kahane, O.C. McGehee, and W. Rudin. I would also like to thank the publisher for the friendly cooperation in the production of this book.
YITZHAK KATZNELSON
Jerusalem
April 1968
The third edition
The second edition was essentially identical with the first, except for the correction of a few misprints. In the current edition some more misprints were corrected, the wording changed in a few places, and some material added: two additional sections in Chapter I and one in Chapter IV; an additional appendix; and a few additional exercises.
The added material does not reflect the progress in the field in the past thirty or forty years. Much of it could and, in retrospect, should have been included in the first edition of the book.
This book was and is intended to serve, as its title makes explicit, as an introduction. It offers what I believe to be the core material and technique, a basis on which much can be built.
The added items in the bibliography expand on parts which are dis-cussed here only briefly (or not at all), and provide a much more up-todate bibliography of Harmonic analysis.
Y.K.
Stanford
June 2003
