Celebratio Mathematica

It is only fit­ting that this con­fer­ence was ori­gin­ally ded­ic­ated to the memory of Ant­oni Zyg­mund. This was the fourth of a series of con­fer­ences de­voted to a branch of Ana­lys­is that was strongly in­flu­enced by him. The series is of­ten re­ferred to as the “El Escori­al” con­fer­ences, since the first three were, in­deed, held in El Escori­al in the province of Mad­rid. For purely lo­gist­ic­al reas­ons this con­fer­ence was held in Mir­a­flores de la Si­erra, also in the province of Mad­rid. Each of these con­fer­ences was or­gan­ized by a group of ana­lysts that was strongly in­flu­enced by the school that Zyg­mund cre­ated. In fact, the first of these con­fer­ences (which took place in 1979) was ded­ic­ated to him and the at­tract­ive poster that an­nounced it has, as its back­ground, the im­age of Ant­oni Zyg­mund. He was proud of this and he kept this poster in his study at home.

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, Mary Weiss, 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 Hardy and Lit­tle­wood on Four­i­er and Tri­go­no­met­ric series, the max­im­al func­tion, sum­mab­il­ity the­ory, the M. Riesz con­vex­ity the­or­em, the study of the prop­er­ties of Le­besgue spaces and their ex­ten­sions, the de­vel­op­ment by F. Riesz 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 that E. M. Stein and 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 Saks, and the text on in­teg­ra­tion, writ­ten with Wheeden, 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 Stélios Pi­chor­ides, the last of Zyg­mund’s stu­dents. This meet­ing is also ded­ic­ated to him. I would like to echo an ob­ser­va­tion made by Jean-Pierre Ka­hane 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.