# Celebratio Mathematica

## Antoni Zygmund

### In Memoriam: Antoni Zygmund(December 26, 1900 – May 30, 1992)

#### by Richard Askey

Ant­oni Zyg­mund died on May 30, 1992, at the age of 91. His math­em­at­ic­al leg­acy lives on, in his pa­pers, his books, and his stu­dents and their stu­dents.

His re­search in­cludes such ma­jor work as the the­ory of sin­gu­lar in­teg­rals in $n$-di­men­sions which goes un­der the name of Calder­ón and Zyg­mund sin­gu­lar in­teg­rals. He was one of the first to real­ize the im­port­ance of com­plex meth­ods in har­mon­ic ana­lys­is and, in­de­pend­ently of Thor­in, dis­covered a com­plex vari­able proof of the M. Riesz in­ter­pol­a­tion the­or­em. He also wrote the first de­tailed treat­ment of Mar­cinkiewicz’s in­ter­pol­a­tion the­or­em. These are the two main tools which lead to the the­ory of in­ter­pol­a­tion spaces, both the com­plex the­ory and the real the­ory. In ap­prox­im­a­tion the­ory, he showed that $L \log L$ is the right sub­sti­tute for $L$ in many ques­tions, and he in­tro­duced the space which he called $\Lambda$ as the right space to sub­sti­tute for $\mathit{Lip}(1)$, and then proved one of the early sat­ur­a­tion the­or­ems. Many of his oth­er con­tri­bu­tions have been used by oth­ers in ap­prox­im­a­tion the­ory.

It is dif­fi­cult for young­er math­em­aticians to real­ize the im­port­ance of the first edi­tion of Zyg­mund’s Tri­go­no­met­ric Series. It was thought at the time when he wrote this book that it would be im­possible to bring or­der in­to the com­plic­ated res­ults which seemed not to have an over­rid­ing struc­ture. I heard this from more than one per­son when I star­ted to read this book, twenty years after it was first pub­lished. After the second edi­tion was pub­lished, I nev­er heard Four­i­er Ana­lys­is de­scribed as a sub­ject without struc­ture, since it was there in these two mar­velous volumes.

As im­port­ant as Zyg­mund’s re­search, his books, and his two joint texts on com­plex and real ana­lys­is were, his work with stu­dents is prob­ably his most im­port­ant leg­acy. The Calder­ón–Zyg­mund sem­in­ar, as it was known when I went to it for two years, was an ex­cit­ing place to learn how math­em­at­ics was done, how prob­lems are un­covered and de­veloped, and what math­em­at­ic­al taste is. There was a one hour lec­ture, with Zyg­mund of­ten mak­ing very per­cept­ive com­ments, and then an in­form­al second hour which was full of open ques­tions, some of which would be solved on the spot by oth­ers in the room. It played a cent­ral role in my de­vel­op­ment as a math­em­atician, and for many oth­ers as well. A list of Zyg­mund’s stu­dents in the United States and their stu­dents is giv­en in Part III of A Cen­tury of Math­em­at­ics in Amer­ica.1 Zyg­mund also had Ph.D. stu­dents in Po­land, in­clud­ing J. Mar­cinkiewicz. Zyg­mund’s ac­count of how Mar­cinkiewicz de­veloped ap­pears in the Col­lec­ted Pa­pers of Mar­cinkiewicz which Zyg­mund ed­ited. The Se­lec­ted Pa­pers of Ant­oni Zyg­mund 2 should be in every ma­jor lib­rary, and the ac­count of Zyg­mund’s work con­tained there should be read by people who want to know about some of the im­pact Zyg­mund had on the de­vel­op­ment of math­em­at­ics in this cen­tury.

Zyg­mund was a gentle man, al­ways look­ing for the best in oth­ers. To quote from the pa­per on his stu­dents,3 Zyg­mund told one of his stu­dents, “Con­cen­trate only on the achieve­ments, and ig­nore the mis­takes. When judging a math­em­atician you should only in­teg­rate $f_{+}$ (the pos­it­ive part of his func­tion) and ig­nore the neg­at­ive part.” This is wise ad­vice from a very wise man.