Edwin Hewitt (1920-1999)

In Memoriam

Memorial from Volume 6, of TopCom

Walter Schempp
Lehrstuhl für Mathematik I
Universität Siegen
D-57068 Siegen

English translation by Glenna F. Burckel

Who was this ever-active, world-travelled, linguistically gifted and lovably extroverted professor of pure mathematics at the University of Washington? Edwin Hewitt was born on the twentieth of January, 1920, in Everett, Washington, U.S.A., the son of a lawyer. After attending school in Everett and Seattle, as well as in St. Louis, Missouri and Ann Arbor, Michigan, with little inclination to devote himself to the study of law like his father, he left the family home at the young age of sixteen, to study mathematics at Harvard University in Cambridge, Massachusetts. With the bestowal of the Ph.D. degree in 1942 his study at Harvard concluded. All his life he felt closely bound to Harvard. Military service in the U.S. Air Force followed, 1943-1945. From 1945-1946, as a Fellow of the John Simon Guggenheim Foundation he was a member of the Institute for Advanced Study in Princeton, New Jersey. After assistant professorships at Bryn Mawr College and the University of Chicago, he returned as a professor of mathematics to the University of Washington in 1954. With the exception of two interruptions, at Yale University in 1959 and at the University of Texas at Austin 1972/73, he held this position until he became emeritus.

Along with mathematics and music Edwin Hewitt loved travel above all. He occupied guest professorships at Uppsala University in Sweden (as "biträdande lärare"), at the Universities of Western Australia and New South Wales, as a Humboldt prize-winner at the University of Erlangen-Nuremberg, at the Steklov Institute in Moscow, as Distinguished Professor at the University of Alaska in Fairbanks, at Hokkaido University in Sapporo, Japan, and at the National University in Singapore. Many of the scientific relationships forged during guest professorships he cultivated through the years. In order to be able to lecture in the language of each host country or in a related language, he learned along with French, also Swedish, Russian, Japanese and German. As he became lonelier he liked to recall how he had played French horn in the university orchestra in Moscow. He used his knowledge of Russian to publish translations of the monographs of M.A. Naimark and A.I. Stern, Theory of Group Representations, and of A.A. Kirillov, Elements of the Theory of Representations in the Grundlehren Series of Springer-Verlag. Despite his marked gift for languages and his untiring readiness to study, he confided in Fairbanks, Alaska, that his knowledge of the language of the Inuit had not progressed very far.

The research area to which Edwin Hewitt had devoted himself was abstract harmonic analysis. In a work about the Fourier transform on compact topological groups dedicated to his fatherly friend Marshall Harvey Stone on the occasion of his retirement in May 1968, Edwin described what he and "the older generation of harmonic analysts" (to which at age 48 he considered himself to belong) wanted understood by abstract harmonic analysis:

"What is the goal of abstract harmonic analysis? One may say that it is to rewrite Antoni Zygmund's monograph for every locally compact Abelian group and every compact non-Abelian group. This is not strictly true, of course: but a major aim is to provide the sort of detailed knowledge about each locally compact Abelian or compact group that we have for the circle T and the line R. Unquestionably some of this can be done. The p-adic numbers, for example, are just as good a group as R, and there is no reason why Hilbert transforms, conjugate functions, Carleson's theorem, Salem's singular measures with small Fourier-Stieltjes transforms and Cantor set supports, et cetera, should not be studied on this group. The same is true of other neo-classical groups, such as the character group of the discrete additive rationals. The classical compact non-Abelian groups are also wide open for detailed analysis. Extremely refined studies of SU(2) are being carried on and others are concerned with detailed analysis on one or another group. The higher-dimensional unitary groups SU(n) for n at least 3 are of great interest to physicists. Here difficulties arise in obtaining explicitly the irreducible unitary representations, although Weyl's work gives an algorithm for computing them all. No reasonable formula is known for decomposing the tensor product of two irreducible representations into irreducible components. But the future looks bright, and for the older generation of harmonic analysts, the main problem is to grasp the new work as quickly as the younger people write it."

"My whole life long I've tried to understand this magic convolution product. I never succeeded."

In the first volume, whose first edition appeared in 1963 and second in 1979, harmonic analysis on locally compact abelian topological groups is developed, with a thorough discussion of the theory of Haar measure and convolution algebras. The existence and uniqueness proof for the quasi-invariant Wiener measure on loop-groups, based on the construction of the Riemann integral, was not known at the time of the second edition, so that those aspects of harmonic and stochastic analysis (based on the Wiener convolution algebra of loop-groups) are not included. Naturally André Weil's famous L'intéegration dans les groupes topologiques et ses applications of 1941 serves as progenitor for the treatment of Pontryagin duality. The plethora of material already available in the sixties thwarted the original goal of writing all harmonic analysis on locally compact abelian topological groups in one manageable volume. An exhaustive treatment would have made more than one monograph necessary.

The monumental second volume appeared in 1970 and treats the representation theory of compact non-abelian groups with the Peter-Weyl theorem as central result: Every compact Lie group is isomorphic to a subgroup of a suitable unitary group. Even today the work is a treasure-trove for every mathematician interested in harmonic analysis, containing many results in this field published for the first time in a textbook. But what about harmonic analysis on topological groups which are neither compact nor locally compact and abelian? For example, the Heisenberg nilpotent Lie group, which is embedded in a central extension of the loop-group of the torus, with its unitary dual and the Fock measure so important in theoretical physics, belongs to this category, so that André Weil's epochal 1964 Acta work "Sur certains groupes d'opérateurs unitaires", about the metaplectic group associated with locally compact abelian topological groups, does not fit into the framework of the second volume.

Edwin Hewitt's name was widely known among pure mathematicians through these exemplarily constructed Abstract Harmonic Analysis volumes. His approach to the harmonic analysis of the "dual objects" of locally compact abelian and compact non-abelian topological groups derived from measure theory and not from the theory of distributions on semisimple Lie groups, nor with a view to applications of topological groups in theoretical physics. Certainly he moved closer to the engineering applications of harmonic analysis in a work written with his son Robert in 1979 about the Gibbs phenomenon of Fourier series, and at the Mathematical Research Institute in Oberwolfach impressively presented their results touching the dawning computer-age as a just-discovered "state secret". However, he felt and thought primarily as a pure mathematician for whom the Fourier transform as a continuous and injective homomorphism of the convolution algebra L¹T into the algebra c₀Z was more congenial then the filterbank property for frequency-phase analysis and for the practical formulation of the uncertainty relation of quantum physics. Considering the work of Harish-Chandra, be became acutely aware of the one-sidedness of his conception of harmonic analysis and its applications, and sometimes he regretted it. He then consoled himself with having climbed some "non-trivial mountains" earlier. Nevertheless, in the course of time his increasing dissatisfaction with the purely structural conception of harmonic analysis and the virtual exclusion of its applications could not be missed.

In addition to his research projects Edwin Hewitt also dedicated himself very conscientiously to teaching. The textbook written with the late Professor Karl R. Stromberg (University of Oregon) on Real and Abstract Analysis, dedicated to the esteemed Marshall H. Stone was a product of his many years of lecturing at the University of Washington.

"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, 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 {\it 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."

It was an experience to listen to Edwin Hewitt and how he tried in lectures on Fourier series of periodic functions to communicate "behind the scenes" to even the less mathematically experienced students the difference between the Dirichlet and the Fejér kernels. Leopold Fejér and above all Marcel Riesz were among the harmonic analysists whose work Edwin Hewitt especially esteemed.

"The big secret I've learned the last few years is to love the students. I wish I'd learned it earlier."

"The only stupid question is the one that isn't asked."

"Exercises are to a mathematician what Czerny is to a pianist."

Retired Professor of Mathematics Practice limited to Analysis