Celebratio Mathematica

I began my gradu­ate stud­ies in math­em­at­ics in the early 1950s and wrote my Ph.D. thes­is un­der the dir­ec­tion of Ant­oni Zyg­mund. The gradu­ate pro­gram offered by the Uni­versity of Chica­go was ex­cel­lent. But I really learned most by at­tend­ing and par­ti­cip­at­ing in the le­gendary “Zyg­mund sem­in­ar”. It was there that I learned the vari­ous top­ics in har­mon­ic ana­lys­is that formed the basis of what is now known as the “Calderón–Zyg­mund School”. Al­berto Calderón had re­ceived his Ph.D. at the Uni­versity of Chica­go be­fore I star­ted my stud­ies with Zyg­mund and had left to join the fac­ulty at Ohio State Uni­versity and, a short time later, at MIT. I felt his pres­ence at the Uni­versity of Chica­go, however, in prac­tic­ally every ses­sion of the Zyg­mund sem­in­ar. He had made im­port­ant con­tri­bu­tions in each top­ic we dis­cussed: in­ter­pol­a­tion of op­er­at­ors, po­ten­tial the­ory and the bound­ary be­ha­vi­or of har­mon­ic func­tions, er­god­ic the­ory, and, of course, sin­gu­lar in­teg­rals. I was sur­roun­ded by not only a re­mark­able fac­ulty but also by a large num­ber of very tal­en­ted gradu­ate stu­dent col­leagues who have by now made im­port­ant con­tri­bu­tions in math­em­at­ics. I could not help, however, hav­ing a feel­ing of great awe for this in­di­vidu­al who was cap­able of mak­ing so many im­port­ant con­tri­bu­tions in such a large num­ber of top­ics. As I stated above, he was an ever present par­ti­cipant of the Zyg­mund sem­in­ar.

I first met Al­berto shortly be­fore I fin­ished my thes­is in the mid-1950s. He vis­ited the Uni­versity of Chica­go, and after present­ing a beau­ti­ful talk on sin­gu­lar in­teg­rals and their con­nec­tion with par­tial dif­fer­en­tial equa­tions, Zyg­mund, the young group that was work­ing with him, and Al­berto met in Zyg­mund’s of­fice and dis­cussed vari­ous top­ics in har­mon­ic ana­lys­is. I re­mem­ber vividly the im­pres­sions Al­berto made on me. He had unique in­sights and ways of look­ing in­to the vari­ous sub­jects we dis­cussed and in a very friendly, open, and gen­er­ous way shared this know­ledge with us. I began to real­ize what was his main strength: he made spe­cial ef­forts to re­duce each concept he con­sidered to clear, simple com­pon­ents, and from this un­der­stand­ing he was able to ar­rive at meth­ods for solv­ing a prob­lem that of­ten were ap­plic­able to many oth­er fields in math­em­at­ics. His con­sid­er­able con­tri­bu­tions in ana­lys­is are be­ing de­scribed else­where in these No­tices; I will not try to give such a de­scrip­tion. I do want to em­phas­ize, however, this im­port­ant fea­ture that is ever present in his re­search: The meth­ods he dis­covered of­ten go way bey­ond the res­ults he ob­tained. Ideas he used in his study of the in­ter­pol­a­tion of op­er­at­ors have had an im­port­ant im­pact in fields that seem totally dif­fer­ent. One of the im­port­ant equa­tions that char­ac­ter­ize wave­lets, for ex­ample, is really a dis­crete ver­sion of what is known as the “Calderón Re­pro­du­cing For­mula”.

Since this first meet­ing I have had sev­er­al op­por­tun­it­ies to be with Al­berto, and on each oc­ca­sion all the feel­ings I de­scribed above were strengthened. I was asked by the math­em­at­ics de­part­ment of the Uni­versity of Chica­go to give the “Zyg­mund lec­tures”, and I presen­ted them in March of this year. Re­cently I be­came in­ter­ested in the math­em­at­ic­al the­ory of wave­lets and chose this top­ic for the three lec­tures I gave. Al­berto at­ten­ded these lec­tures and ex­pressed an in­terest in the sub­ject; in par­tic­u­lar, he asked me to send him what I had writ­ten. I was very ex­cited at the pro­spect of es­tab­lish­ing an area of com­mon in­terest with him. I ob­tained a copy of a book on wave­lets I had re­cently coau­thored with Eu­genio Hernández and put it to­geth­er with a col­lec­tion of pa­pers I thought might in­terest Al­berto, but then I heard the sad news of his passing away. Along with many oth­ers, I will miss him very much.