1.1. Review by: Hazleton Mirkil.
The American Mathematical Monthly 67 (1) (1960), 93-94.

Here are four distinct survey articles bound together in one volume. They differ in content, and even more in style. ... Hewitt's Survey of harmonic Analysis should be read together with the article by Mackey (Bull. Amer. Math. Soc. 56 (1950), 385-412). The Hewitt article gives a more or less complete catalogue of theorems proved up to 1956 about function spaces defined over locally compact abelian groups. But Hewitt's point of view is uncompromisingly abstract, and he therefore has a right to ignore results that depend on special properties of (say) the reals or the integers.

2.1. From the Publisher.

The book is based on courses given by E. Hewitt at the University of Washington and the University of Uppsala. The book is intended to be readable by students who have had basic graduate courses in real analysis, set-theoretic topology, and algebra. That is, the reader should know elementary set theory, set-theoretic topology, measure theory, and algebra. The book begins with preliminaries in notation and terminology, group theory, and topology. It continues with elements of the theory of topological groups, the integration on locally compact spaces, and invariant functionals. The book concludes with convolutions and group representations, and characters and duality of locally compact Abelian groups.

2.2. From the Preface.

When we accepted the kind invitation of Prof Dr F K Schmidt to write a monograph on abstract harmonic analysis for the Grundlehren der Mathematischen Wissenschaften series, we intended to write all that we could find out about the subject in a text of about 600 printed pages. We intended that our book should be accessible to beginners, and we hoped to make it useful to specialists as well. These aims proved to be mutually inconsistent. Hence the present volume comprises only half of the projected work. It gives all of the structure of topological groups needed for harmonic analysis as it is known to us; it treats integration on locally compact groups in detail; it contains an introduction to the theory of group representations. In the second volume we will treat harmonic analysis on compact groups and locally compact Abelian groups, in considerable detail.

The book is based on courses given by E Hewitt at the University of Washington and the University of Uppsala, although naturally the material of these courses has been enormously expanded to meet the needs of a formal monograph. Like the other treatments of harmonic analysis that have appeared since 1940, the book is a lineal descendant of A Weil's fundamental treatise. The debt of all workers in the field to Weil's work is well known and enormous. We have also borrowed freely from Loomis's treatment of the subject, from Naimark, and most especially from Pontryagin. In our exposition of the structure of locally compact Abelian groups and of the Pontryagin-van Kampen duality theorem, we have been strongly influenced by Pontryagin's treatment. We hope to have justified the writing of yet another treatise on abstract harmonic analysis by taking up recent work, by writing out the details of every important construction and theorem, and by including a large number of concrete examples and facts not available in other textbooks.

The book is intended to be readable by students who have had basic graduate courses in real analysis, set-theoretic topology, and algebra as given in United States universities at the present day. That is, we suppose that the reader knows elementary set theory, set-theoretic topology, measure theory, and algebra. ...

In our effort to make the book useful for specialists, we have included a large corpus of material which a beginning reader may find it wise to omit. ...

Many sections also contain historical notes We have tried to trace the history of the principal theorems and concepts, but we obviously have not produced a complete history, and it would be foolish to claim that we have produced one correct in every detail Also, while some of the results we give are new so far as we know, failure to quote a reference for a given theorem should not be construed as a claim of originality on our part.

For the reader's convenience, we have assembled in three appendices certain ancillary material not easily accessible elsewhere, which is essential for one part or another of the theory. These appendices may be read as the reader encounters references to them m the main text.
...
The reading and research by both of us on which the book is based have been generously supported by the National Science Foundation, U.S.A., and by a fellowship of the John Simon Guggenheim Memorial Foundation granted to E Hewitt. We extend our sincere thanks to the authorities of these foundations, without whose aid our work could not have been done.

Finally, we record our gratitude to Prof Dr F K Schmidt for his original suggestion that the book be written, and to the publishers for their dispatch, skill, and attention to our every wish in producing the book.

2.3. Review by: John Hunter Williamson.
Proceedings Edinburgh Mathematical Society 13 (4) (1963), 345-346.

This is the first of two volumes, in which the authors set out to give an account of abstract harmonic analysis as at present understood - that is, the generalisations of Fourier series and integrals to the case of locally compact abelian groups, and of the classical theory of matrix representations of finite groups to representation of locally compact topological groups by operators on a Hilbert space. The present volume comprises (to quote from the authors' Preface) " ... all of the structure of topological groups needed for harmonic analysis ... integration on locally compact groups in detail ... an introduction to the theory of group representations". The second volume will treat "harmonic analysis on compact groups and locally compact abelian groups, in considerable detail". The scope of volume one is indicated by the chapter headings: Preliminaries, Elements of the theory of topological groups, Integration on locally compact spaces, Invariant functionals, Convolutions and group representations, Characters and duality of locally compact abelian groups. There are three appendices, totalling 53 pages (Abelian groups, Topological linear spaces, and Introduction to normed algebras), and a 14-page bibliography. It would be a considerable over-simplification, but perhaps not entirely misleading, to say that the present volume goes as far as the Pontryagin duality theorem in the abelian case, and the Gelfand-Raikov theorem (on the sufficiency of the irreducible unitary representations) in the general case. The duality theorem is here treated by the original structure-theoretic method of Pontryagin and van Kampen; the alternative (and shorter) Fourier transform proof is promised for volume two. The core of the book is, of course, in the last two chapters; the rest is essentially preparatory, though there is much of independent interest to be found in the earlier chapters.

The treatment is designed to be suitable for beginners as well as experts; in this context a beginner is assumed to be a graduate with a good knowledge of algebra, topology and measure theory - equivalent, say, to substantial parts of van der Waerden, Kelley and Halmos. The exposition is careful and detailed, and indications are given of those sections that may be omitted on a first reading of the subject. An uninitiated reader is likely to find his difficulties of a global rather than a local nature; the total mass of material is rather formidable. Harmonic analysis is, in its essentials, a rather more elementary subject than appears from the present volume. It is possible to penetrate to the basic theorems without the use of quite such a formidable array of analytical apparatus; in particular, only a rather primitive version of measure theory is needed. However, with suitable guidance, this book could very well be used as a text by a beginning research student.

For the more experienced analyst, there is much of interest; the wealth of detail and the numerous indications of related work make the book very useful as a reference. The historical and bibliographical notes, and the "miscellaneous theorems and examples" that appear after most of the sections are particularly valuable. The writing has evidently been done with great care ... This is clearly destined to be a standard reference for many years to come, and should be on the shelves of every library and on the desk of every harmonic analyst. It is to be hoped that the second volume will appear without undue delay, to complete a major addition to the literature of the subject.

2.4. Review by: Fred E J Linton.
The American Mathematical Monthly 73 (3) (1966), 331.

The book under review is the first half of a projected two volume monograph on abstract harmonic analysis. The second volume (we are told in the preface) "will treat harmonic analysis on compact groups and locally compact Abelian groups, in considerable detail." The present volume "gives all of the structure of topological groups needed for harmonic analysis as it is known to us; it treats integration on locally compact groups in detail; it contains an introduction to the theory of group representations." Three appendices recall relevant facts about Abelian groups, topological linear spaces, and normed algebras. In a sense, this volume lays out the lubricants required for the smooth operation of the machinery to be found in the second. One could equally well maintain that this volume lays the foundation on which the edifice of the second is to rest. In either case, it seems indisputable that the absence of much serious representation theory in the present volume and the presence of so much material of prerequisite character will disappoint a fair number of readers. Equally indisputable, however, and much to the book's credit, is the comprehensive treatment of the subjects that are covered. There are many exercises in sections entitled Miscellaneous theorems and examples, and copious historical notes following most paragraphs. The presence of an index of symbols is a thoughtful service.

2.5. Review by: C Terry C Wall.
The Mathematical Gazette 49 (368) (1965), 235-236.

Harmonic analysis generalises Fourier series and integrals to functions on locally compact abelian topological groups, and generalises representation theory for finite groups over the complex numbers to representations of arbitrary locally compact topological groups. ...

One distinctive feature of this book is its size: this is explained partly by the large amount of introductory material which is included, and partly by a certain attempt at completeness-in the main sections of the book, and also in the "miscellaneous theorems and examples" at the end of most chapters, which have been carefully chosen as illustrative examples for the results, counterexamples to show the hypotheses in the main theorems to be necessary, and other results in the literature not included elsewhere in the book. I was also pleased to note that considerable attention is paid to the purely algebraic, and to the purely topological properties of the groups (particularly in the sections on abelian groups) - the authors are not committed to any one approach; in particular they do not always use epsilontics when a verbal argument would do as well (they sometimes do - e.g. pp. 367-8). Of course, the compensating disadvantage of the amount of detail is that it is some- times hard to see the wood for the trees, particularly for anyone new to the subject. Some suggestions for a first reading are given in the preface. The book is carefully written; the proofs are all detailed and easy to follow, and there appear to be no mistakes or misprints!

2.6. Review by: Leopoldo Nachbin.
Bull. Amer. Math. Soc. 73 (1967), 292-294.

For a certain category of mathematicians, integration theory is, or at least used to be, restricted to its charming virtues on $\mathbb{R}^{n}$. Measures however provide us with a powerful and elegant tool when used along with the intuition that keeps fertile company to them in a large quantity of algebraic, analytic and geometrical situations.
...
In the present volume, the authors give the structure of topological groups needed for harmonic analysis; treat integration on locally compact groups; and introduce the readers to group representations. They promise formally that a second volume will go into the intimate life of compact groups and locally compact Abelian groups, in considerable detail. The authors "hope to have justified the writing of yet another treatise on abstract harmonic analysis by taking up recent work, by writing out the details of every important construction and theorem, and by including a large number of concrete examples and facts not available in other textbooks." The pleasant, useful and important features of this volume, however, go far beyond those claimed by the authors. The book is intended to be readable by students having a first year graduate training, and to be useful for specialists as well. As such it happens, although it may not have been meant, to be a twin brother with a different temperament of Walter Rudin's Fourier analysis on groups, 1962.
...
The authors did a very nice job in writing a book which strictly speaking is not exclusively a classroom text and is not either solely a reference treatise, but shares both features as a sort of a two-bladed knife in the good sense of the expression.

The above description of chapters is not a fair hint of the richness and completeness of the book within the ample boundaries set forth by the authors for the students and the specialists they aimed at.

Many interesting sections are devoted to the history of the major theorems and ideas, to illuminating examples, as well as to the partisanship that fortunately enough authors are still entitled to follow.

3.1. From the Preface of Volume 2.

This book is a continuation of Volume I of the same title. We constantly cite definitions and results from Volume 1. The textbook Real and abstract analysis by E Hewitt and K R Stromberg [Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965], which appeared between the publication of the two volumes of this work, contains many standard facts from analysis. We use this book as a convenient reference for such facts, and denote it in the text by RAAA. Most readers will have only occasional need actually to read in RAAA.

Our goal in this volume is to present the most important parts of harmonic analysis on compact groups and on locally compact Abelian groups. We deal with general locally compact groups only where they are the natural setting for what we are considering, or where one or another group provides a useful counterexample.
...
Obviously we have not been able to cover all of harmonic analysis. The field, already immense, is growing rapidly at the present day. We were limited by space, by time, by our own abilities. We have presented the parts of the subject that every harmonic analyst must know: representation of compact groups; the Weyl-Peter theorem; Plancherel's theorem; Wiener's Tauberian theorem. Beyond this, we have been guided largely by personal inclination. As the writing progressed, one question led naturally to another.

3.2. Review by: Alessandro Figà-Talamanca.
Bull Amer. Math. Soc. 78 (2) (1972), 172-178.

Abstract harmonic analysis is concerned with the theory of Fourier series and integrals in the context of topological groups.

As would be expected even from such a summary description "abstract" theories and "classical" theories are intimately connected. Abstract harmonic analysis cannot replace classical Fourier analysis but it is now almost impossible to work in the latter subject without having in mind the "abstract" developments. The relationship between classical theories and new theories however is by no means clear.
...

The book under review can also be considered as an attempt to provide the basic tools to do research in harmonic analysis well beyond the boundaries of locally compact abelian groups; and the class of nonabelian groups which the authors choose to emphasise is that of compact groups. Of course the two volumes contain an incredible amount of information on general topological groups as well and they present, in an original fashion and in full detail, many results and techniques which are proper of harmonic analysis on locally compact abelian groups; and indeed the first volume contains a very detailed analysis of the structure of locally compact abelian groups with a wealth of illuminating examples.

This reviewer feels however that perhaps the most important contribution of the second volume of this treatise is that of opening a wide avenue for research in harmonic analysis on noncommutative compact groups.

Notwithstanding the great number of research papers which are published on this subject, compact noncommutative groups have not been in the limelight of harmonic analysis. Research on this topic has been rather fragmented and without a definite and clear direction. It is to be expected that the publication of this book will alter this situation considerably and will also induce many students of commutative harmonic analysis to work on noncommutative groups. It is probable in any case that many people will be surprised by the extent to which harmonic analysis on noncommutative compact groups is developed in the book under review. It is not surprising that the methodic and intelligent application of techniques of "abstract analysis" should provide generalisations of results from the theory of Fourier series. But the book under review offers to the reader a number of specific tools which are typical of harmonic analysis such as the constructions of special kernels and special trigonometric polynomials and the technique of randomising the coefficient of a Fourier series. In fact what might come as a surprise to many is that some of these tools can be constructed for all compact groups, often by performing the necessary calculations only in the case of particular compact subgroups of the unitary group.

This treatise however is by no means a monograph on compact groups. Its encyclopaedic character is especially evident in the first volume which, in the authors' words, "gives all of the structure of topological groups needed for harmonic analysis as it is known to us; it treats integration on locally compact groups in detail; it contains an introduction to the theory of group representations."

4.1. From the Preface.

This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present customarily offered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have included every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Hence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner, we have given elementary avatars of all important theorems, with appropriate suggestions for skipping. We have given complete definitions, explanations, and proofs throughout, so that the book should be usable for individual study as well as for a course text.
...
Modern analysis draws on at least five disciplines. First. to explore measure theory, and even the structure of the real number system, one must use powerful machinery from the abstract theory of sets. Second, as hinted above, algebraic ideas and techniques are illuminating and sometimes essential in studying problems in analysis. Third, set-theoretic topology is needed in constructing and studying measures. Fourth, the theory of topological linear spaces ["functional analysis"] can often be applied to obtain fundamental results in analysis. with surprisingly little effort. Finally, analysis really is analysis. We think that handling inequalities, computing with actual functions, and obtaining actual numbers, is indispensable to the training of every mathematician. All five of these subjects thus find a place in our book. To make the book useful to probabilists, statisticians, physicists, chemists, and engineers, we have included many "applied" topics: Hermite functions; Fourier series and integrals, including Plancherel's theorem and pointwise summability; the strong law of large numbers; a thorough discussion of complex-valued measures on the line. Such applications of the abstract theory are also vital to the pure mathematician who wants to know where his subject came from and also where it may be going.

With only a few exceptions, everything in the book has been taught by at least one of us at least once in our real variables courses, at the Universities of Oregon and Washington. As it stands, however, the book is undoubtedly too long to be covered in toto in a one-year course. We offer the following road map for the instructor or individual reader who wants to get to the centre of the subject without pursuing byways, even interesting ones. ...

4.2. Review by: John A Erdos.
Quarterly of Applied Mathematics 24 (4) (1967), 395.

In this book the authors have given an account of the essential topics for a rigorous course in advanced analysis. The book also contains some more advanced material and many applications of the theory. Chapter One deals with set theory, including the axiom of choice, cardinal and ordinal numbers, and a construction of the real and complex number fields. There follows a chapter on topology and a detailed treatment of spaces of continuous functions. Integration theory is the subject of Chapter Three. The account given starts with the Riemann-Stieltjes integral and culminates, through the Daniell extension procedure, with the full generality of integration on arbitrary measure spaces. Chapter Four is an introduction to functional analysis giving the basic theorems on Banach and Hilbert spaces, together with applications to Fourier transforms and special functions. In Chapter Five there is an account of differentiation including the Lebesgue-Radon-Nikodyn theorem and several applications. Finally there is a chapter devoted to integration on product spaces.

The authors are to be congratulated on their choice of material for this book. The presentation is clear and complete and the more peripheral sections are all interesting and often contain illuminating comments. While the book as a whole would be too much for all but the best students, with some omissions, an excellent course could be given from it. Suggestions as to what might be left out are given in the preface. The presence of many applications, both in the text and in exercises, should make this a most suitable book for prospective applied mathematicians.

The index of some fourteen pages should help in the use of this work as a reference. However for this it is necessary to be familiar with the notation, because special symbols, once introduced, are used without further reminders. The production leaves something to be desired. Too often the pages have a rather cramped appearance, frequently the same symbols are printed on the same line in different sizes and fractional exponents are in disproportionately large type. However these are minor blemishes in an excellent book.

4.3. Review by: Deborah Tepper Haimo.
The American Mathematical Monthly 75 (1) (1968), 94-95.

The theory of functions of a real variable is presented from the point of view that modern analysis, though largely abstract, has its roots in classical, concrete mathematics. In addition to a general and comprehensive development of the traditional tools of real function theory, topics of interest in contemporary research are included.

The book contains six chapters, the first two of which deal with the fundamentals of the subject. These begin with an informal review of the elements of set theory and with the construction of the real and the complex number fields, continue with the development of set theoretic topology as it applies to analysis and with the study of spaces of continuous functions, and conclude with a presentation of several versions of the Stone- Weierstrass theorem. In Chapter 3, the Riemann-Stieltjes integral is introduced, and by a very general extension, using the Darboux-Daniell method, the Lebesgue integral is derived and its properties explored, the chapter culminating with the Riesz representation theorem. Chapter 4 is devoted to a study of general Banach spaces, including the function spaces $L^{p}$ and their conjugate spaces, and Hilbert spaces. The next chapter begins with a classical treatment of the theory of differentiation and continues with a characterization of those functions which are indefinite integrals. An extension of the concept of indefinite integral leads to the Lebesgue-Radon-Nikodym theorem which is proved under very general conditions. This generality is relaxed in the study of product spaces in Chapter 6, where Fubini's theorem, for example, is established on the assumption of $\sigma$-finiteness of the measure spaces.

The book gives a careful, thorough, and well organised presentation of differentiation and integration. It is replete with interesting and helpful remarks stressing the highlights of a theorem, emphasising the importance of given hypotheses, indicating the place of a result in the general structure, or pointing out different directions in which the theory has developed. Although no bibliography is included, references and historical notes are contained either in the body of the text or in the footnotes. There are numerous exercises of varying degree of difficulty and many provocative illustrative examples.

The book is an excellent source of reference for the professional, but as a text for first or second year graduate students, it offers far too much material, as the authors themselves indicate. In sketching an outline of an abbreviated course of study, however, even they find it difficult to make up their minds to limit the topics to be covered. For example, in the preface, the instructions given for the selection of sections to be taken up in Chapters 4 and 5 are at variance with the introductory remarks at the beginnings of these chapters. Further, for students without adequate mathematical maturity, the generality of the treatment may prove difficult, but the enthusiastic and challenging approach cannot fail to generate interest and curiosity.

4.4. Review by: John Charles Burkill.
The Mathematical Gazette 51 (378) (1967), 365-366.

By tradition, the Theory of Functions of a Real Variable, from the pen of (say) Lebesgue, Hobson or Carathéodory, covered limits, differentiation and integration, in greater depth than a course of analysis even a cours d'analyse. The variables really were real numbers.

With the advance of knowledge in the last three or more decades, the domain of integration has evolved from the Euclidean line and plane into general sets which are equipped with measures. The book of Halmos (1950) was followed by many good accounts of Measure and Integral, for instance those of Zaanen, Munroe, Berberian and now Kingman and Taylor.

Measure and Integration is the backbone of the book of Hewitt and Stromberg, and the inclusion of Abstract Analysis in the title proclaims the generality of the setting. They discuss certain topics more fully than most writers of a book on Measure and/or Integration would. The foundation chapters on Set Theory and Topology are more systematic and more valuable. Moreover the authors rightly say "Finally, analysis really is analysis." The general is made special, and oases are revealed in the desert. We are given admirable accounts of Hermite functions, Fourier series, Cesaro sums, the Hardy-Littlewood maximal theorem etc.

The prose is lively and gives a necessary aeration to the symbolism which is a surfeit of script and Gothic letters, suffixes, suffixes sub- suffixes, brackets within brackets, bars. ...

Briefly, the book is excellent and at present it seems to be unique. It earns praise for being both compact and complete. Most of it is for the graduate only. Such a wealth of material in 460 pages cannot be absorbed without hard work, but his reward will be great.