A Trip from Classical to Abstract Fourier Analysis Kenneth A. Ross lassical Fourier analysis began with Question. Which functions h in L1(G) can be writ- Fourier series, i.e., the study of periodic ten as f ∗ g? That is, which h in L1(G) factors as functions on the real line R. Because the convolution of two L1-functions? trigonometric functions are involved, Theorem ([32, Raphaël Salem, 1939]). L1(T) ∗ we will focus on 2π-periodic functions, L1(T) = L1(T). Also L1(T) ∗ C(T) = C(T), where C 2 which are determined by functions on [0; 2π). C(T) is the space of continuous functions on T. This is a group under addition modulo 2π, and this group is isomorphic to the circle group What year did Salem publish this result? Accord- T = fz 2 C : jzj = 1g under the map t ! eit . ing to Zygmund’s book [37] and Salem’s published Classical results can be viewed as results on the works [33, page 90], this paper was published in compact abelian group T, and we will do so. 1939. According to Hewitt & Ross [17], this paper Similarly, Fourier analysis on Rn can be viewed was published in 1945. The paper was reviewed in as analysis on the locally compact1abelian (LCA) March 1940, so 1939 is surely correct. The story is group Rn. more complicated than a simple error in [17]. The Our general setting will be a locally compact journal had received papers for that issue in 1939, but no more issues were published until 1945 group G. Every such group has a (left) translation- because of World War II. Unfortunately, Hewitt and invariant measure, called Haar measure because I used the publication date on the actual journal. Alfred Haar proved this statement in 1932. For Who was Salem? He was a very talented banker G = Rn, this is Lebesgue measure. Haar measure in France who did mathematics on the side. When for G = T is also Lebesgue measure, either on World War II broke out, he went to Canada and [0; 2π) or transferred to the circle group T. Haar ended up a professor at M.I.T. Antoni Zygmund measure will always be the underlying measure gives a brief account of Salem’s life in the Preface to in our Lp(G) spaces. In particular, L1(G) is the Salem’s complete works (Oeuvres Mathèmatiques) space of all integrable functions on G. Functions 1 [33]. Edwin Hewitt told me that Salem was a real in L (G) can be convolved: gentleman who treated graduate students well. Z 3 f ∗ g(x) = f (x + y)g(−y) dy Salem died in 1963. 4 1 n G In 1957, Walter Rudin announced that L (R )∗ 1 n 1 n or L (R ) = L (R ). This prompted the eminent Z French mathematician and leading co-author of f (xy)g(y−1) dy for f ; g 2 L1(G). Nicholas Bourbaki, Jean Dieudonné, to write to G 2 Under this convolution, L1(G) is a Banach algebra. See the Appendix for more about Salem’s theorems. 3Starting in 1968, the Salem Prize has been awarded almost every year to a young mathematician judged to have done Kenneth A. Ross is emeritus professor of mathematics at outstanding work in Fourier series, very broadly interpreted. the University of Oregon. His email address is rossmath@ The prize is considered highly prestigious and many recipi- pacinfo.com. ents have also been awarded the Fields Medal later in their 1For us, locally compact groups are always Hausdorff. careers. DOI: http://dx.doi.org/10.1090/noti1161 4Bull. Amer. Math. Soc. abstract 731t, p. 382. 1032 Notices of the AMS Volume 61, Number 9 Walter Rudin, on December 17, 1957, as follows Here is Dieudonné’s response of January 17, [30, Chapter 23]: 1958: Dear Professor Rudin: Dear Professor Rudin: In the last issue of the Bulletin of the Thank you for pointing out my error; as AMS, I see that you announce in abstract it is of a very common type, I suppose I 731t, p. 382, that in the algebra L1(Rn), any should have been able to detect it myself, element is the convolution of two elements but you know how hard it is to see one’s of that algebra. I am rather amazed at that own mistakes, when you have once become statement, for a few years ago I had made convinced that some result must be true!! a simple remark which seemed to me to Your proof is very ingenious; I hope you disprove your theorem [9].5 I reproduce the will be able to generalize that result to proof for your convenience: arbitrary locally compact abelian groups, Suppose f ; g are in L1 and ≥ 0, and but I suppose this would require a somewhat for each n consider the usual “truncated” different type of proof. With my congratulations for your nice functions f = inf(f ; n) and g = inf(g; n); n n result and my best thanks, I am f (resp. g) is the limit of the increasing Sincerely yours, J. Dieudonné sequence (fn) (resp. (gn)), hence, by the usual Lebesgue convergence theorem, h = In fact, Rudin proved: f ∗g is a.e. the limit of hn = fn ∗gn, which is Theorem (Walter Rudin [27, 1957], [28, 1958]). obviously an increasing sequence. Moreover, L1(G) ∗ L1(G) = L1(G) for G abelian and locally 2 fn and gn are both in L , hence it is well Euclidean. In particular, this equality holds for R n known that hn can be taken continuous and and R . bounded. It follows that h is a.e. equal to I have heard stories that Rudin heard Salem or a Baire function of the first class. However, Zygmund talk about this theorem in Chicago and it is well known that there are integrable then ran off and stole the result for himself. But functions which do not have that property, the timing and context are all wrong, and Rudin’s and therefore they cannot be convolutions. proof is quite different from Salem’s. When I wrote I am unable to find any flaw in that to Rudin in 1979, I did not of course allude to the argument, and if you can do so, I would very stories I’d heard. Nevertheless, his response began: much appreciate if you can tell me where I First of all, let me say that I regret very much am wrong. that I never got around to acknowledging Sincerely yours, J. Dieudonné Salem’s priority in print. I was unaware As Rudin notes, there are nonnegative L1- of his factoring—when I did my stuff, and functions that cannot be written as f ∗ g where later there never seemed to be a good f ; g are nonnegative.6 Rudin wrote to Dieudonné opportunity to do anything about it. explaining this subtle error. In fact, in a separate letter to me in 1979, now lost, Rudin commented, The Cohen Factorization Theorem “It’s a very subtle error, and I would certainly have Walter Rudin did his work at the University of believed it if I hadn’t proved the opposite.” Later, Rochester. His first book, his very successful in his memoir [30], Rudin wrote, “Needless to undergraduate analysis text [26], fondly known say, I was totally amazed. Here was Dieudonné, a as “baby Rudin” or “blue Rudin,” was published world-class mathematician and one of the founders in 1953 and lists Rudin as at Rochester. However, of Bourbaki, not telling me, a young upstart, ‘you he wrote this book while at M.I.T. In 1957, Paul are wrong, because here is what I proved a few Cohen went to the University of Rochester without years ago’ but asking me, instead, to tell him what finishing his Ph.D. at the University of Chicago he had done wrong! Actually, it took me a while under Antoni Zygmund. Neither Zygmund nor to find the error, and if I had not proved earlier Cohen were impressed with his thesis, but Rudin that convolution-factorization is always possible and others finally persuaded Cohen to write up his in L1, I would have accepted his conclusion with thesis and return to Chicago for his final exam. no hesitation, not because he was famous, but About this time, Rudin showed Cohen his work because his argument was simple and perfectly on factorization. Cohen saw Féjer and Riesz kernels correct, as far as it went.” and the like and said, “aha, approximate identities.” An approximate unit is a sequence or net feαg 5The remark was in a footnote, and the footnote includes the such that x = limα xeα = limα eαx for all x in the following assertion: “Il suffit pour le montrer de considérer topological algebra. Soon Cohen found an elegant, le cas où f et g sont positives;” completely elementary (using only the definitions) 6See the preceding footnote. short proof of: October 2014 Notices of the AMS 1033 Theorem ([5, Paul J. Cohen, 1959]). If A is a Banach Hewitt’s main application was: algebra with a bounded approximate unit, then Theorem ([15, Edwin Hewitt, 1964]). For 1 ≤ p < AA = A, i.e., each element of A can be factored as 1, we have L1(G) ∗ Lp(G) = Lp(G) for any locally a product of two elements in A.7 compact group G. Paul J. Cohen was a brilliant mathematician who went on to do substantial work in my field, Philip Curtis and Alessandro Figà-Talamanca harmonic analysis, for a couple of years.