According to Yitzhak Katznelson (*An Introduction to Harmonic Analysis*, p. vii), “Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups.” This is a pretty vague definition, and covers a lot of ground.

In the simplest case, if *f* is a periodic function of one real variable, say of period 2π, then we can think of *f* as being defined on the circle group — that is, the additive group **T** of real numbers modulo integer multiples of 2π. It is a basic fact (whose proof Steven Krantz reproduces on pp. 44–46 of the book under review) that the *characters* of this group — that is, its continuous complex-valued homomorphisms — are just the exponentials *x* ↦ e^{inx}, where *n* is an integer; in other words, what amounts to the same thing, sines and cosines. Given any periodic function *f*, we can try to express it in terms of these particularly nice functions. This attempt leads to the subject of Fourier series. There are many generalizations. The group **T** can be replaced by the real line, by higher-dimensional Euclidean space, or (eventually) by any locally compact group. In the case of Euclidean space, the symmetries include not only translations (that is, actions by elements of the group), but also dilations and rotations; these special symmetries play an important role in the structure of harmonic analysis in this particular setting.

A “panorama,” according to the Oxford English Dictionary, is “a complete and comprehensive survey or presentation of a subject”. (The *OED* entry is worth looking up; the word was coined only in the year 1789.) Steven Krantz’s book *A Panorama of Harmonic Analysis*, the heftiest volume so far in the MAA Carus Monograph series, does not, of course, give a “panorama” in this sense. Indeed, it would be impossible to write a complete and comprehensive presentation of such a vast subject as harmonic analysis. Krantz has chosen to give a very selective treatment of Fourier analysis in Euclidean space, starting with the most basic problems, but leading ultimately to quite recent developments. As is well-known, one aspect of the subject is its wealth of applications to science and engineering, as well as to other parts of mathematics. An account of many of these applications can be found in T. W. Körner’s book *Fourier Analysis*. Although Krantz mentions the importance of applications, and even gives a list of some of them, his book is focused almost entirely on “pure” Fourier analysis, as the following sketch of its contents will make clear.

Chapter 0 gives a very brief treatment of some of the basic ideas of measure theory and functional analysis which are needed as a foundation. The last section of this chapter states and proves two “fundamental principles of functional analysis” which are applied many times in the sequel; Krantz denotes them by “FAPI” and “FAPII”. The first, FAPI, is a sort of (much more elementary) converse to the Banach-Steinhaus theorem; Krantz uses it to prove norm-convergence of Fourier series in various settings. The second, FAPII, is an analogous result based on the concept of “weak boundedness”. It is one of the elements in Carleson’s proof of the almost-everywhere pointwise convergence of the Fourier series of an *L*^{2} function (see Hunt’s survey article “Developments related to the a.e. convergence of Fourier series”, pp. 26–27), but does not seem to have received a distinctive name. Krantz uses it to prove pointwise convergence of Fourier series for various summability methods.

Krantz’s Chapter 1 begins by tracing the genesis of Fourier series back to the 18th-century controversies about the vibrating string. Following Fourier, Krantz then derives the heat equation for an insulated rod. (This derivation is ostensibly based on the three “physical principles” listed on p. 35. Unmentioned, as usual, is the physically implausible assumption that the rod itself neither expands nor contracts as a result of the flow of heat in it; see Truesdell, *Tragicomical History of Thermodynamics, 1822-1854*, p. 70.) There follows Fourier’s solution of the heat equation by separation of variables. A separate section gives a derivation of the wave equation, making the usual facile assumptions of smallness. (The historical material in this chapter seems to be based primarily on Langer’s Slaught Memorial Paper, which, in turn, appears to depend entirely on secondary sources.)

Against this background, Krantz introduces the concept of Fourier series, and proves that the Fourier series of an integrable function *f* converges to *f* at every point at which *f* is differentiable. (Krantz notes that this is a “not-terribly-well-known” theorem.) Proving convergence of Fourier series under less restrictive hypotheses than differentiability is difficult, and there are various other approaches one could take. One is to introduce “summability methods”, which replace the partial sums of the Fourier series by suitable averages. Krantz studies, in particular, the summability methods of Fejér and Poisson. In another direction, instead of considering pointwise convergence, we could work with the topology of an appropriate function space, typically *L*^{p} space. Of course, these approaches can also be combined: thus we can ask whether the Fejér or Poisson sums converge in the *L*^{p} norm. The remainder of Chapter 1 takes up these questions. The main tactic here is to bring in the apparatus of functional analysis.

Specifically, the partial sums produced by summability methods can be represented by certain integral operators, in fact convolution operators, whose kernels are “summability kernels”. Furthermore, the convergence of these methods in norm, for functions *f* belonging to *L*^{p}(**T**), 1 ≤ p ≤ ∞, comes from the fact that the associated convolution operators are *bounded* operators on *L*^{p}. To handle the corresponding question for *pointwise* convergence, Krantz introduces the Hardy-Littlewood maximal operator,

and shows that it satisfies a “weak boundedness” property. Finally, questions of norm-convergence for ordinary partial sums are shown to depend in a similar way on the boundedness of another integral operator, the Hilbert transform.

Thus, this first chapter introduces the theme that questions about Fourier analysis can be understood by studying certain integral operators, using techniques of functional analysis. The remainder of the book consists essentially of a further development of these ideas.

Chapter 2 develops the basic properties of the Fourier transform, including the Heisenberg Uncertainty Principle. Chapter 3 returns to Fourier series, now in higher dimensions. Following an article by Ash, Krantz shows that the question of how to define the “partial sums” of a multiple series is quite delicate. For example, we could take the sum of all terms corresponding to lattice points which lie in a ball of radius *R*, then take the limit as *R* → ∞. That is called “spherical summation”. Or, the ball could be replaced by a rectangular box of some fixed shape. But one can find examples of series whose spherical sums converge, but whose rectangular sums diverge, or vice versa. Once again, these questions of convergence can be related to notions of functional analysis — in particular, the notion of a “Fourier multiplier”. The machinery of the Fourier transform is essential here; the connection between Fourier transforms and Fourier series makes use of the dilations of Euclidean space. Among the various ways of forming partial sums, the method of spherical summation is the most difficult to study — as it turns out, because the sphere has infinitely many distinct tangent planes. The last section of Chapter 3 presents Fefferman’s remarkable discovery that the problem of spherical convergence of double Fourier series is related to the classical “Kakeya needle problem”. The problem, posed in 1917 by the Japanese mathematician S. Kakeya, is this. Dip a needle in ink and lay it on a piece of paper. Now slide the needle on the paper in such a way that its endpoints get reversed. What is the minimal area that must be smeared with ink? Besicovitch found in 1928, amazingly, that there is no minimum! The smeared area can be made arbitrarily small.

The notion of multiple Fourier series makes sense for a function defined on a torus — that is, a product of circles. There is another way to generalize the one-dimensional concept, namely, to functions defined on a higher-dimensional sphere — a subspace invariant under rotations. This generalization leads to the concept of spherical harmonics, the subject of Chapter 4. Here Krantz has followed very closely the exposition given in the book *Introduction to Fourier Analysis on Euclidean Spaces* by Stein and Weiss. (In order to compute the dimension of the space of spherical harmonics, Stein and Weiss, pp. 138–139, rederive the standard formula for the number of combinations of *n* things taken *r* at a time with repetitions allowed, a formula which goes back, at least, to Jacob Bernoulli’s *Ars Conjectandi* of 1713. Krantz, pp. 174–175, faithfully copies this derivation, giving no hint that it is a well-known fact. Perhaps mathematicians with different specialties should talk to one another more often.)

Chapter 5 takes up the theme of the Hilbert transform from Chapter 1, now seen as the simplest example of a “singular integral”. Krantz first introduces the fractional-integral operators of M. Riesz, then the higher-dimensional singular integrals of Calderón-Zygmund. He shows how these singular integrals can be used to give a real-variable definition of the Hardy spaces *H*^{p}, classically defined as spaces of holomorphic functions in the unit disk satisfying norm bounds analogous to those of the Lebesgue spaces *L*^{p}. The remainder of the chapter considers various ways of thinking about these new spaces.

In order to prove the fundamental theorem of Calderón-Zygmund on the *L*^{p}-boundedness of singular-integral operators, Krantz proceeds, in Chapter 6, in a more abstract setting. It turns out that the essential arguments are measure-theoretic, rather than Fourier-analytic. The fundamental notion here is that of a “space of homogeneous type”. Such a space is, first of all, a “quasi-metric space”. A quasi-metric space is like a metric space, except that the triangle inequality has been generalized to the inequality

where ρ is the quasi-metric and *C* is a constant (depending on the space). To get a space of homogeneous type, you start with a quasi-metric space and impose on it a regular Borel measure which respects the quasi-metric in an appropriate way. An example of a space of homogeneous type is a smooth boundary of an open domain in Euclidean space, the measure being Hausdorff measure.

One can now define an analogue of the Hardy-Littlewood maximal operator on any space of homogeneous type and prove the corresponding weak-boundedness property by arguments which are completely analogous to those which were used in Chapter 1. One consequence is a generalization of the classical Lebesgue differentiation theorem. Krantz also sets up an abstract theory of Calderón-Zygmund singular-integral operators in this context, and proves the theorem which generalizes and subsumes the Calderón-Zygmund *L*^{p}-boundedness theorem. Krantz concludes the chapter by describing (but not proving) the “T(1)” theorem of David-Journé, 1984, which makes it possible to prove *L*^{2}-boundedness of certain operators which (unlike convolution operators) are not translation-invariant.

The final chapter, Chapter 7, sketches another recent development in harmonic analysis, the theory of wavelets, mainly following a *Monthly* article by Robert Strichartz. Of course, the literature on wavelets is enormous. For some time, there has been an argument among historians of mathematics about whether or not there are “revolutions” in mathematics. Though Krantz does not use the word “revolution,” it seems to me that his point here is that wavelets are not a “revolution,” but simply a further development of the basic principles of harmonic analysis.

The book concludes with a sequence of appendices covering further background material, notably the interpolation theorems of Riesz-Thorin and Marcinkiewicz, which are in fact among the fundamental tools used in the book. It is not entirely clear to me what principle was applied in order to decide which material went into Chapter 0, and which into the appendices.

According to the dust jacket, the book is intended for “graduate students, advanced undergraduates, mathematicians”. Although Krantz states that “prerequisites are few”, it is clear that the reader should be comfortable with measure theory and functional analysis. Krantz claims that the review of these subjects given in Chapter 0 is enough to enable even the “complete neophyte” to understand the rest of the book. I don’t buy this. By the time we have reached Chapter 6, for example, we are working with product-measures defined on an abstract space. At another point (p. 211), Krantz appeals to the Banach-Alaoglu theorem on the weak-* compactness of the unit ball in the dual of a Banach space. (To be sure, this occurs in a section which is considered “optional”.) I do agree with Krantz that the book provides good motivation for learning about these subjects, but I think that a reader who has not already studied measure theory and functional analysis will have to do considerable reading on the side in order not to lose the thread. The interpolation theorems of Riesz-Thorin and Marcinkiewicz don’t seem to be standard material yet, although there is a nice treatment in the real-analysis textbook by Folland, which Krantz takes as one of his basic references. The theory of Schwartz distributions keeps trying to make an appearance, but Krantz resolutely pushes it aside. At some points, a knowledge of complex function theory would also be helpful. There are no undergraduates at my school who have all this background, though I concede that some might exist somewhere. I think that the book would be valuable mainly for graduate students who are looking for a comparatively quick introduction to recent work in harmonic analysis.

Traditionally, math books (going back to Euclid, I guess) have consisted of lists of theorems with proofs. Such a treatment is convenient if you want to check the details of the proofs, but it may not give you much notion of the intuitive basis of the subject. It seems to me that, to get a good understanding of any branch of mathematics, both sides are necessary. Without an intuitive understanding and an overall view, the student cannot see how to fit all the individual details together; but, without studying rigorous proofs, the student may not be able to grasp the hard technicalities that constrain the intuitive notions. The trouble is that it is hard to present both sides in a single work.

Krantz has chosen to emphasize the intuitive side of the subject. Very often, his proofs are only sketched, and measure-theoretic (or distribution-theoretic) details are usually omitted. This is certainly a reasonable choice. Despite some welcome developments in the literature over the last few decades, there is still a shortage of good treatments of this kind.

In my opinion, however, a good intuitive presentation requires more care and attention to detail than a “theorem-proof” approach. For the latter, it suffices just to write down all the steps; but for the former, accuracy and precision are essential if the student is not to be left feeling lost. If a theorem is stated imprecisely in a “theorem-proof” account, the student will probably be able to determine from the proof what the correct statement should be. But if the details of the proof aren’t there, the student is at the author’s mercy.

Unfortunately, Krantz shows many signs of carelessness. Thus, his statement of Fubini’s theorem (p. 8) is wrong, even though restricted to continuous integrands. You can’t always interchange the order of integration in an iterated improper integral, even if the integrand is bounded and continuous, as simple and well-known examples show (*e.g.*, Apostol, *Mathematical Analysis*, p. 455, Ex. 14-14). Of course, you also need to know, for example, that the integrand is absolutely integrable, a hypothesis that Krantz slips in, without comment, further down the page. After developing the Fourier inversion formula under the assumption that both *f* and its transform are continuous (pp. 111–112), Krantz observes that it follows (assuming *f* and its transform are both *L*^{1}) that *f* is equal almost everywhere to a continuous function. This is a paralogism. After having defined the Fourier transform for *L*^{1} functions and for *L*^{2} functions, Krantz uses the Riesz-Thorin theorem to conclude (p. 116) that it can be defined for *L*^{p} functions with 1 ≤ p ≤ 2, but notes that this doesn’t work for *p* > 2; yet in his later treatment of Fourier multipliers (*e.g.*, p. 132), he manipulates the Fourier transform of an *L*^{p} function with *p* > 2 without any further comment. Of course, for *p* > 2, the Fourier transform can be defined as a Schwartz distribution (Katznelson, *Harmonic Analysis*, pp. 146-154), something Krantz avoids discussing. It seems to me that this kind of carelessness puts roadblocks in the way of the graduate student who is trying to master this material. To be sure, the student who manages to work through these difficulties will probably have gained something from the experience; the question is, whether this is the most efficient use of the student’s time.

Aside from Chapter 4, on spherical harmonics, and Chapter 7, on wavelets, I would say that the main theme of the book is to prove the boundedness of various linear operators on *L*^{p} spaces. Krantz has exposed some very powerful techniques for doing this. One might ask, however, why we are so interested in proving that these operators are bounded. It doesn’t seem to me that this is a question we are interested in purely for its own sake (unless, perhaps, we have become specialists); we want boundedness as a tool for solving other problems. Well, all right, Krantz has given us an example of how *L*^{p}-boundedness can be applied: namely, to prove the convergence of Fourier series.

But why are we interested in the convergence of Fourier series? Physicists and engineers, of course, are not interested. To them, examples of continuous functions whose Fourier series diverge on sets of measure zero, or Kolmogorov’s example of an *L*^{1} function whose Fourier series diverges everywhere, are mere monstrosities. To mathematicians, on the other hand, the question of convergence is certainly important. It is essential to clarify the sense in which a Fourier series can be said to represent a function. But here again, I would say, the question of convergence is not our main interest. We can admire the elegance and ingenuity of the convergence proofs, but, ultimately, the reason we want to know that the series converge is so that we can apply them to solve other problems.

The difficulty here, it seems to me, is that Krantz’s exposition has built up this great machinery of singular integrals on spaces of homogeneous type, but hasn’t shown us what can be done with it. As far as we can tell from this book, the only application appears to be to proving convergence of Fourier series, and, to me, that’s not enough. Motivation is important, but there must also be a payoff. Chekhov said that if you bring a cannon on stage, you must fire it. Surely there are applications to partial-differential equations, probability theory, even number theory. Show us what some of these applications are. Perhaps the chapter on wavelets, which adds little to what is readily available in the literature, could have been omitted, and the space saved used for discussing applications of some of the general theorems. (For that matter, there are applications to wavelets, which Krantz has alluded to briefly on p. 308.) Of course, applications often require further background of their own, but, here at least, it is not necessary to provide all the details. What the student needs is an idea of where the theory is going.

Krantz has adopted a format for making bibliographical references which, regrettably, in my opinion, has become very popular in the mathematical community in recent years. He assigns to each item in the bibliography a code, consisting of three or more capital letters. For example, the treatise of Dunford and Schwartz gets the code [DUS]. There doesn’t seem to be any particular rule as to how the code for a particular item is constructed. It is usually formed from some combination of letters from the authors’ last names, but not always. For example, the book on Fourier series in Euclidean space by Elias Stein and Guido Weiss has the code [STG]. Perhaps this particular code was chosen in order that the codes not occur out of alphabetical order in the bibliography itself, something that frequently occurs when this method is adopted. There are two separate items in Krantz’s bibliography having the same code, [LAN]. I find this method of giving references opaque and unpleasant to use. Since the codes are constructed arbitrarily, I can never guess what book or article they stand for, and, after having looked the reference up, I can never remember what the code means the next time I see it. It seems to me that it would be much more helpful to the reader to give the authors’ last names and the date of publication: for example, “Dunford-Schwartz, 1958”. I counted 200-some references in Krantz’s book, about one every 2 pages. I estimate that giving references in the longer form would add less than 2 pages to this 370-page book.

In Krantz’s book *A Primer of Mathematical Writing*, p. 76, he instructs us: “do *not* give in-text biblographic references that have the form `see Dunford and Schwartz’ (for those not in the know, [DS] is a three volume work totaling more than 2500 pages). The only correct and thorough way to give a reference is to cite the specific theorem or the specific page.” In the *Panorama*, Krantz has neglected his own precept; none of his bibliographic references gives a page number. In particular (p. 18), after noting that “the analogue of the Hahn-Banach theorem for linear operators *L*:X → Y is false”, he advises the student to look in Dunford & Schwartz for help! (One way in which the analogue of the Hahn-Banach theorem fails is that a closed subspace of a Banach space need not have a topological complement; thus, the identity operator on the subspace cannot be extended to a projection from the whole space to the subspace. For an example of such a subspace, see Whitley, “Projecting *m* onto *c*_{0}”. I haven’t been able to find where, if at all, this question is discussed in Dunford & Schwartz, though there is a brief mention on pp. 553-554. There is a thorough discussion in Nachbin, “Some problems in extending and lifting continuous linear transformations”.)

Krantz has written a big book, an exciting, challenging, confusing book. He has included a tremendous amount of material. I enjoyed reading the book, and I learned a lot about harmonic analysis that I didn’t know. A well-prepared graduate student who is willing to work pretty hard should find the book to be a valuable introduction to recent ideas in harmonic analysis; the student could then fill in the details by going on to study the works of Elias Stein.

References:

- A few other books on Harmonic Analysis:
Yitzhak Katznelson, *An Introduction to Harmonic Analysis*, Wiley, 1968; Dover reprint, 1976, ISBN: 0-486-63331-4.

T. W. Körner, *Fourier Analysis*, Cambridge University Press, 1988, ISBN: 0-521-25120-6.

Elias Stein, *Singular Integrals and Differentiability Properties of Functions*, Princeton University Press, 1970, ISBN: 0-691-08079-8.

Elias Stein and Guido Weiss, *Introduction to Fourier Analysis on Euclidean Spaces*, Princeton University Press, 1971, ISBN: 0-691-08078-X.

Elias Stein, *Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals*, Princeton University Press, 1993, ISBN: 0-691-03216-5.

- Dunford & Schwartz:
Nelson Dunford and Jacob T. Schwartz, *Linear Operators*, 3 vols., Wiley-Interscience, 1958, 1963, 1971, ISBN: 0-470-22605-6, 0-470-22638-2, 0-471-22639-4.

- On the Hahn-Banach theorem:
Robert Whitley, “Projecting *m* onto *c*_{0}”, *American Mathematical Monthly*, **73**, 1966, pp. 285-286.

Leopoldo Nachbin, “Some problems in extending and lifting continuous linear transformations”, *Proceedings of the International Symposium on Linear Spaces*, Jerusalem, 1960; Jerusalem Academic Press, 1961, pp. 340-350.

- Hunt’s survey paper:
Richard A. Hunt, “Developments related to the a.e. convergence of Fourier series”, *Studies in Harmonic Analysis*, ed. J. M. Ash, *MAA Studies in Mathematics*, Mathematical Association of America, 1976 (ISBN: 0-88385-113-X), pp. 20-37.

- Multiple trigonometric series:
J. Marshall Ash, “Multiple trigonometric series”, *Studies in Harmonic Analysis*, ed. J. M. Ash, *MAA Studies in Mathematics*, Mathematical Association of America, 1976 (ISBN: 0-88385-113-X), pp. 76-96.

Charles Fefferman, “The multiplier problem for the ball”, *Annals of Mathematics*, **94**, 1971, pp. 330-336.

- Langer’s Slaught Memorial Paper:
R. E. Langer, “Fourier’s Series: The Genesis and Evolution of a Theory”, Herbert Ellsworth Slaught Memorial Paper I, *American Mathematical Monthly*, **54**, 1947.

- A thorough discussion of the vibrating-string controversy (and much else), with extensive quotations from the original sources, can be found in:
C. Truesdell, *The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788*; *Leonhardi Euleri Opera Omnia*, Series II, vol. 11, part 2, 1960.

- For an analysis and critique of Fourier’s derivation of the heat equation:
C. Truesdell, *The Tragicomical History of Thermodynamics, 1822-1854*, Springer, 1980, pp. 55-78, ISBN: 0-387-90403-4.

- On the derivation of the wave equation, see
Stuart S. Antman, “The equations for large vibrations of strings”, *American Mathematical Monthly*, **87**, 1980, pp. 359-370.

- Besicovitch’s solution of the Kakeya problem:
A. S. Besicovitch, “The Kakeya problem”, *American Mathematical Monthly*, **70**, 1963, pp. 697-706.

- Jacob Bernoulli’s derivation of the number of combinations can be found in
David Eugene Smith, *A Source Book in Mathematics*, Dover, 1959, pp. 272-277.

- Textbooks on real analysis:
Tom M. Apostol, *Mathematical Analysis*, Addison-Wesley, 1957.

Gerald B. Folland, *Real Analysis: Modern techniques and their applications*, Wiley, 1984, ISBN: 0-471-80958-6. This book contains all the background needed for the study of Krantz’s *Panorama*.

- An intuitive introduction to Schwartz distributions and their relation to harmonic analysis:
Robert Strichartz, *A Guide to Distribution Theory and Fourier Transforms*, CRC Press, 1994, ISBN: 0-8493-8273-4. Krantz’s proof of the Uncertainty Principle is taken essentially from pp. 126-129.

- Strichartz’s article on wavelets:
Robert S. Strichartz, “How to make wavelets”, *American Mathematical Monthly*, **100**, 1993, pp. 539-556.

- Krantz on mathematical writing:
Steven G. Krantz, *A Primer of Mathematical Writing*, American Mathematical Society, 1997, ISBN: 0-8218-0635-1.

- The actual quote from Chekhov is, “One can’t put a loaded gun on the stage if no one plans to fire it.” From a letter to Lazarev-Gruzinsky, 1889; see Peter M. Bitsilli,
*Chekhov’s Art: A stylistic analysis*, Ardis, 1983 (ISBN: 0-88233-489-1), p. 42.

Stacy G. Langton ([email protected]) is Professor of Mathematics and Computer Science at the University of San Diego. During his recent sabbatical, he translated three articles by Euler.