During these past years much has been written about Zygmund and this is not the only conference that has been held in his honor. I do not want to repeat what has been said in the various articles and dedications involved in this homage. Perhaps it is appropriate to cite the article published in 1989 by the American Mathematical Society in Part III of A Century of Mathematics in America, pages 343–368. This article is entitled “The School of Antoni Zygmund” and from it we can learn some aspects of his character and the immense influence he had in the world of mathematics. One of the features of this article is that it lists two generations of students: his and those who were trained by his students. This list contains 179 names, and, of course, this number is still growing. The Selected Papers of Antoni Zygmund, published by Kluwer Academic Publishers in 1989, presents another good picture of the influence Zygmund had on the development of analysis in the twentieth century. In this article I would like to complement what appears elsewhere and cite some more personal memories I have of this kind, gentle and most influential man. Hopefully, then, the various articles of Zygmund that have appeared in recent years will, when put together, give a reasonably complete and proper image of the man. The articles mentioned above give a good accounting of Zygmund’s research and his success with students and collaborators. Perhaps not enough has been said about his books and what they mean to mathematics. I will concentrate most of my remarks around their importance and how they were regarded by Zygmund.
Zygmund was finishing the second edition of his book on trigonometric series when I was working on my Ph.D. thesis as one of his students. Shortly after we obtained our degrees, he asked me and my wife,, who was also his student, to proofread the galleys of this book. At that time (during the academic year 1958–59) we were young and energetic and we did a very thorough and careful job of proofreading. Influenced by this reading and the legendary Zygmund seminar, we felt that we knew reasonably well what the subject called “Trigonometric series” was all about. It is interesting to muse upon how much we take for granted in everyday life; we push a button and a television screen lights up and scenes from the other side of the globe are displayed. As we do this we do not think about all the scientific, engineering, and intellectual effort that made it possible. We just make use of these results. Much the same can be said about reading Zygmund’s second edition. It was, of course, considerably harder to do this proofreading than it is to push a button; but having done so we went ahead and used the results we learned without giving much thought to how all this material was organized into a meaningful whole. I have to confess that I have read very few mathematics books from cover to cover; the first edition of Zygmund’s book Trigonometrical Series is another member of this small set. I do not think that I am unusual among my colleagues. Mathematicians are not scholars in the sense this term is used in other disciplines. They do make use of libraries, books, and journals. Most often they do this in order to examine a very particular result and will focus only on the material that is closest to this result. I think that they are remarkably good in this endeavor and this practice does, indeed, advance the development of new mathematics. This activity, however, makes it difficult to obtain a “broad view” of a subject. Zygmund’s two books Trigonometrical Series and Trigonometric Series represent a remarkable achievement of bringing together several different areas of Mathematics into one coherent whole. The titles of these books are misleading because of their unpretentious simplicity. In his first book, Zygmund gave a structure to many apparently disconnected results and techniques: the considerable work of and on Fourier and Trigonometric series, the maximal function, summability theory, the convexity theorem, the study of the properties of Lebesgue spaces and their extensions, the development by of what are known as Hardy spaces, probabilistic methods for studying trigonometric series, convergence results, and differentiability of integrals are just some of the topics that are introduced in Trigonometrical Series and then updated and refined in Trigonometric Series.
Zygmund was very much aware of the effort he put into the writing of these books. As mentioned in the A.M.S. article cited above, he complained that the writing of Trigonometric Series cost him at least thirty research papers. I was present when J. E. Littlewood, in a visit to the University of Chicago, told him: “many generations of mathematicians will thank you.” He was also aware that Littlewood was right. A little known fact is that he told Mary and me after we finished our proofreading: “Now that the second edition is finished I have started working on the third,” and he showed us extensive notes of expositions of the development of Singular Integrals and other higher dimensional aspects of Fourier Analysis. Each of us who worked with him in the later fifties and early sixties was aware of his conviction that the future of Fourier Analysis lay in “higher dimensions and other settings associated with higher dimensional Euclidean spaces.” He also encouraged his students to carry on his program. This was one of the motivations thatand I had for dedicating to him our book Introduction to Fourier Analysis on Euclidean Spaces.
As mentioned in some of the other articles that have been written recently in memory of Zygmund, he wrote other books. His book on the theory of functions, written with, and the text on integration, written with , are well known and highly regarded. Not well known is that he wrote a book on differential equations (in Polish), mostly devoted to ordinary differential equations. I could not find a reference to it in any of the lists of his publications; thus, this book may never have been published. I had the good fortune of taking a course he gave on this subject. It was based on this book and he did show me the manuscript he was using. I present this as evidence that he devoted considerable time and energy on thinking how mathematics, at all levels, should be presented and motivated. He had strong opinions on this subject and I learned a lot about this aspect of my profession from him as well.
His gentleness, his generosity, his broad interest (in politics and literature, as well mathematics), and his sense of humor have been well described in other articles. I cannot ignore, in this article, the terrible tragedy that we experienced during this meeting in Miraflores: the sudden death of, the last of Zygmund’s students. This meeting is also dedicated to him. I would like to echo an observation made by in the article he wrote about Stélios that also appears in these proceedings. To those of you who did not know Zygmund personally, but did have a chance to get to know Stélios, let me say that the latter presented an image that was similar to that of the former. It is remarkable how these two individuals shared the same gentleness, generosity, general interest, and sense of humor.