Ed is well known as a topologist and as an analyst. In fact, much of his work concerned the structure of and analysis on topological algebraic structures like locally compact abelian groups (LCA groups) and topological semigroups. My friend Wis Comfort will report on his work in general topology in an article in a future issue of TopCom. In the early 1950s, Ed turned his attention to various aspects of analysis.
One of the most widely referenced results is the Hewitt-Savage Zero-One Law in probability [1955, Theorem 11.3]. It was clear immediately that this was an important result which would appear in most modern probability books. In that paper, they wrote that this result
"was commented on by several who saw a prepublication copy of this paper. Blackwell, and Chung and Derman wrote us independently that they had become interested in the following question in connection with forthcoming publications. Is it true that the partial sums of a sequence of identically distributed independent random variables visit an arbitrary Borel set infinitely often with probability either 0 or 1? As they point out, the affirmative answer, which they had already demonstrated in certain cases, is an immediate consequence of Theorem 11.3. Halmos and Doob have shown us direct proofs, both of which make it plain that the theorem is close to and scarcely deeper than the ordinary 0-1 law. These proofs are, with their authors' permission, presented below."The main purpose of this fine paper was, in Ed's own words [HF]: "a construction of measures on extreme points of a convex set that is a special case of what later became Choquet theory."
Another early paper that Ed was very pleased with was about finitely additive measures and written jointly with Kosaku Yosida [1952]. Ed so greatly admired Yosida that he insisted that Yosida be the first-listed author. Ed later decided this was a bad policy, even when he worked with other alphabetically-challenged co-authors whom he greatly admired, like Herbert Zuckerman. Ed's note [1957b] with John Williamson played an important role in my life because it was the first paper I tried to read and understand. Hewitt had told me to study it, and I spent what seemed like months on this short note. I wore out my copy, found some trivial generalization, and finally drifted to a more promising topic.
Ed wrote about a dozen papers with his colleague Herbert Zuckerman. Ed wrote [HF]: "I found in Zuckerman a wonderful friend and collaborator. I learned most of my classical mathematics from him, a side that I had foolishly neglected, having been lured to follow the sirens of functional analysis and set-topology. Herbert and I worked together from the autumn of 1948 until the day of his tragic, untimely death in June 1970. I am proud that my name stands alongside his, attached to now classical theorems in Diophantine approximation, the structure of semigroups, and abstract harmonic analysis." Their first program was to study abstract convolution algebras, usually over semigroups. Thus the program was to do abstract harmonic analysis for various classes of semigroups, which were often discrete and usually commutative, though one paper [1957c] is devoted to finding all irreducible representations of a certain semigroup Tn containing the symmetric group Sn. These papers involved careful analysis of semigroups and techniques from number theory. It's clear from these papers that both authors enjoyed intricate computations.
In the meantime, Ed was also making contributions to harmonic analysis on LCA groups. This subject became organized in his mind when he wrote one-fourth of the book [1958a]. His contribution was titled, A survey of abstract harmonic analysis; the other authors for this volume were Irving Kaplansky, Marshall Hall, Jr., and Robert Fortet. It was this survey that led to the contract with Springer-Verlag for the book Abstract Harmonic Analysis, which in turn became a two-volume affair.
Let me tell you more about those books. In the spring of 1959, Ed visited Yale for a special term on harmonic analysis; other visitors included Walter Rudin, Philip C. Curtis, Jr., and Bertram Yood, and I'm sure I am inadvertently omitting other important people. And, of course, Shizuo Kakutani was there. I managed to tag along. I was a favored graduate student, but without a thesis yet. However, Hewitt knew me from our close work on real variables class notes. One day that spring, Ed walked up to me and asked me if I'd like to work with him on his harmonic analysis book. My reward would be that I'd be an editorial assistant; his model was the book at Yale: Dunford & Schwartz, for which Bade and Bartle were editorial assistants. I could see right off that a bigger reward would be that I'd learn a lot, so I didn't hesitate to accept. I was a good editorial assistant. Hewitt had the vision for the book and knew the literature very well. He initially wrote most of the sections. I wrote two sections (out of 26) and the three appendices. My main task was to make sure that everything was correct and consistent, be sure the notation was best possible, look for better proofs (I succeeded occasionally), etc. We were both happy with the arrangement, and I would have been happy to be known as his editorial assistant. But one day Ed generously said, "You did half the work, you should get half the credit and be co-author." Clearly that generosity changed my life!
As happens, the harmonic analysis book project became more and more ambitious, and a second volume "on harmonic analysis on compact groups and on locally compact Abelian groups" was promised. However, Ed was not exaggerating when he noted [HF]: "It was seldom easy to work with me," so he decided that he would do Volume II by himself. Eventually, we decided to do the project together and this time I was truly an equal co-author. Roughly speaking, we each initially wrote half of the book and then mercilessly criticized each other until we were happy with the product. Overall, both projects were enjoyable, but I enjoyed working on Volume II more. Incidentally, Ed talked about writing a third volume, devoted entirely to the measure algebra M(G) of an LCA group G. When the fine book by Colin Graham and Carruth McGehee [GM] came out, I told people that they could view it as Volume III. Hewitt was not amused.
Hewitt and Zuckerman continued to work together throughout the 1960s, and they focused more and more on LCA groups. One key paper was the pioneer article [1959a] on lacunary Fourier series. They carried forward applications of functional analysis that had been noted by Banach. Rudin's lucid article [Ru] codified the theory by establishing attractive terminology (Sidon sets, \Lambdap-sets, etc.), giving clear proofs, and posing several problems. In [1970e] we built on Rudin's treatment in the setting of not-necessarily-abelian compact groups. Later Hewitt and I, alone and in joint work with Robert E. Edwards, wrote several papers on lacunarity and other topics. Indeed, there is at least one paper written by each non-empty subset of {Edwards, Hewitt, Ross}. The three papers written by all three of us were done before Edwards and I met. This was more unusual in the 1970s, at which time the complications of co-authorship grew exponentially with the number of authors. Edwards brought to the projects powerful techniques from functional analysis as well as substantial knowledge about Fourier analysis. The Edwards-Hewitt paper [1965] produced a complete analogue of Féjer-Lebesgue pointwise summability for Fourier and Fourier-Stieltjes transforms on arbitrary LCA or compact non-Abelian groups. It played a key role in volume II of Abstract Harmonic Analysis [1970e] and has been widely cited.
Hewitt and Zuckerman did some very interesting work on singular measures on compact Abelian and LCA groups. The idea of "dissociate sets" was well exploited; these are sets that are independent enough to make them lacunary; the model is {3n : n in N}. One of these papers, [1966], includes a construction of singular, continuous, probability measures \mu on a nondiscrete LCA group G so that \mu * \mu has these properties (P): it is absolutely continuous with respect to Haar measure and, via the Radon-Nikodym theorem, represents a function in Lp(G) for all p < \infty. For compact G, Karl Stromberg [St] generalized this to obtain an uncountable collection F of linearly independent probability measures so that each \mu * \nu has the properties (P), \mu, \nu in F.
Hewitt also wrote two fine, technically challenging, papers with Kakutani on the subject. Their goal was to understand better the multiplicative linear functionals on the measure algebra M(G) of an LCA group G. This is a fascinating topic, because this measure algebra is much more complicated than L1(G) whose multiplicative linear functionals are all given by characters on G. As Ed noted, "one of these (papers) includes the first construction of what are now called Kronecker sets." In the 1970s, in collaboration with Gavin Brown, Hewitt returned to the study of singular measures on LCA groups with a detailed study of these measures on the circle group and real line. In the same decade, Ed worked with Gunter Ritter and made a detailed study in three papers of the growth and decrease of Fourier transforms.
Ed's paper [1963b] shows the potential power of a harmless-looking footnote. You don't need to know the result to appreciate footnote 5: "We conjecture that the number 2\aleph1 can be replaced in this theorem by 2c without recourse to the continuum hypothesis. Our methods do not yield this result, however, and to obtain it would apparently require some quite delicate facts about the structure of compact Abelian groups having dense subgroups that are continuous isomorphic images of Ra." Ed was absolutely right and three authors independently gave essentially the same proof; see [Ra], [Ro] and [Va]. Clearly there should not have been three papers on this minor result, but there were various communication problems among the authors.
In 1959, Paul J. Cohen [Co] gave a beautiful elementary proof of a factorization theorem for Banach algebras with a bounded approximate unit. It implied, for example, that L1(G) * L1(G) = L1(G) for any locally compact group G, and before Cohen the result was known only for some LCA groups. Cohen, with an example, showed that he clearly knew that the result also applied to what are now called Banach modules, so one could argue that all the ideas were in Cohen's paper. But the wide range of interesting applications motivated many authors to mine the field. In particular, Hewitt [1964a], and independently Phil Curtis and Figà-Talamanca [CF], showed that L1(G) * Lp(G) = Lp(G) for 1 <= p < \infty, and much more.
A typical "maximum problem" arises from some general inequality like the Hausdorff-Young inequality, ||f^||q <= ||f||p (under suitable hypotheses), for which it is natural to ask for the extremal functions, i.e., the functions for which equality holds. In a beautiful paper, Hewitt and Hirschman [1954b] characterized such maximal functions for LCA groups. Compact abelian groups have lots of maximal functions, but for some groups, like the real line, only the zero function is a maximal function. Using some methods of Hirschman, Ed and I [1968a] generalized this result to not-necessarily-abelian compact groups. The result for the real line raised the question of whether the norm of the operator f to f^ might be less than 1 and, if so, what those norms are. This question was substantially answered in the very interesting paper by Beckner [Be]; the norms are less than 1 and vary with p.
Let me mention two other papers that Ed and I wrote. Our paper [1965b], written in Seattle in the summer of 1964, was great fun but does not seem to have led to further mathematics. The paper [1974a] on rearrangements of Fourier series has a more interesting history. In the late 1950s Hewitt wanted to generalize some old results of Hardy and Littlewood to all compact abelian groups, but only had partial results. He talked about them at Harvard and maybe other places in the spring of 1959 (when we were visiting Yale), after which he abandoned the problem. But he kept the notes and I resurrected the project when I visited him in Texas in the spring of 1973. Together we obtained the general result he wanted. This paper has motivated more recent work; see [Gu1] and [Gu2].
Like many of us, Ed's true love was hard classical analysis. This is illustrated in work with Gavin Brown [1984a], where they construct a large new class of everywhere positive trigonometric series, and in several papers written with Gunter Ritter concerning Fourier series and conjugate Fourier series on solenoids, i.e., the dual groups of arbitrary noncyclic subgroups of the additive group of rational numbers, endowed with the discrete topology.
In a completely different area, and in collaboration with the chemist George D. Halsey, Jr., Ed contributed to the mathematical interpretation of the 12-tone system in music that was originated by Arnold Schoenberg. According to Ed, they published a "major work in the subject," [1978]. I am unqualified to hazard an opinion. I have an English translation that I will gladly share. Halsey and Hewitt also wrote a Monthly article [1972b] on the superparticular ratios in music.
Ed Hewitt wrote his share of interesting expository articles. Among them are two Monthly articles, published in 1960. One is a sales pitch for compactness [1960], but it gets mighty technical for a Monthly article. The other paper [1960b] provides an integration-by-parts formula for Stieltjes integrals. A somewhat friendlier version, in the setting of Riemann-Stieltjes integrals, can be found in my book [Ro1, 35.19]. Also, he wrote a very nice expository article on the Gibbs phenomenon with Robert Hewitt (no relation) [1979], which I feel deserved wider recognition.
Ed Hewitt is also well known for the now-classical text Real and Abstract Analysis [1965c], which he wrote with Karl Stromberg. It is still a very useful reference. Finally, I think Ed would like me to mention that he translated five books from the Russian, one in collaboration with his younger daughter Lise (Elizabeth). His translation of Natanson's Theory of Functions of a Real Variable was one of the best real analysis books available in the 1950s.
I've only touched on a small part of Ed's work and only mentioned some of his fine collaborators. His collaborations were very important to him, and he must have found them especially rewarding. My last joint paper with Ed was published in 1974. After that we drifted apart, and I missed out on a lot of fun. Here is a list of Ed's co-authors: Carl B. Allendoerfer; M. Ja. Antonovskij; Nakhlé Asmar; Gavin Brown; David Chudnovsky; Gregory Chudnovsky; Robert E. Edwards; Irving Glicksberg; George D. Halsey; Robert E. Hewitt; Isidore I. Hirschman, Jr.; Shizuo Kakutani; Yitzhak Katznelson; Shozo Koshi; Dusa McDuff; Gunter Ritter; Kenneth A. Ross; Herman Rubin; Leonard J. Savage; Karl R. Stromberg; Yuji Takahashi; Eugene P. Wigner; John H. Williamson; Kosaku Yosida; Herbert S. Zuckerman.
[Be] William Beckner Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159-182. MR52 #6317
[CF] Philip C. Curtis, Jr. and Alessandro Figà-Talamanca, Factorization theorems for Banach algebras, Function algebras, pp. 169-185. Scott-Foresman and Company 1966. MR34 #3350
[Co] Paul J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-205. MR21 #3729
[HF] Edwin Hewitt, So far so good - my life up to now, Proceedings of the HewittFest, May 6-7, 1988, pp. 1-7 (privately published). Reprinted in Math. Intelligencer 12 (1990), 58-63. MR91j:01046
[GM] Colin C. Graham and O. Carruth McGehee, Essays in Commutative Harmonic Analysis, Springer-Verlag 1979. MR81d:43001
[Gu1] Archil Gulisashvili, Rearrangements of functions on a locally compact abelian group and integrability of the Fourier transform, J. Functional Anal. 146 (1997), 62-115.
[Gu2] Archil Gulisashvili, Rearrangement-invariant spaces of functions on LCA groups, J. Functional Anal. 156 (1998), 384-410.
[Ra] M. Rajagopalan, Characters of locally compact abelian groups, Math. Zeitschr. 86 (1964), 268-272. MR31 #4852
[Ro] Kenneth A. Ross, Locally compact groups and the continuum hypothesis, Colloquium Math. 13 (1964), 21-25. MR30 #2103
[Ro1] Kenneth A. Ross, Elementary analysis: the theory of calculus, Springer-Verlag 1980. MR81a:26001
[Ru] Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227. MR22 #6972
[St] Karl R. Stromberg, Large families of singular measures having absolutely continuous convolution squares, Proc. Cambridge Philos. Soc. 64 (1968), 1015-1022. MR37 #6693
[Va] N. Th. Varopoulos, A theorem on cardinal numbers associated with a locally compact Abelian group, Proc. Camb. Phil. Soc. 60 (1964), 701-704. MR29 #3570
Kenneth A. Ross
Department of Mathematics
University of Oregon
Eugene, OR 97403-1222
ross@math.uoregon.edu
fax: 541-346-0987