Celebratio Mathematica

Antoni Zygmund

The school of Antoni Zygmund

by Ronald R. Coifman and Robert S. Strichartz

To most math­em­aticians, the words “har­mon­ic ana­lys­is” bring to mind a nar­row sub­field of ana­lys­is ded­ic­ated to very tech­nic­al and clas­sic­al sub­jects in­volving Four­i­er series and in­teg­rals. In fact, it is a very broad field that draws from, in­spires, and uni­fies many dis­cip­lines: real ana­lys­is, com­plex ana­lys­is, func­tion­al ana­lys­is, dif­fer­en­tial equa­tions, dif­fer­en­tial geo­metry, to­po­lo­gic­al groups, prob­ab­il­ity the­ory, the the­ory of spe­cial func­tions, num­ber the­ory, ….  Sev­er­al math­em­aticians have con­trib­uted to the breadth and in­flu­ence of har­mon­ic ana­lys­is. We men­tion only a few names of those who were act­ive in this cen­tury be­fore the second World War: Bern­stein, Be­sicov­itch, Boch­ner, Bo­hr, Den­joy, Fe­jér, Hardy, Kaczmarz, Kolmogorov, Le­besgue, Lit­tle­wood, Lus­in, Menschov, Pa­ley, Plancher­el, Pless­ner, Privalov, Rademach­er, F. and M. Riesz, Stein­haus, Szegö, Titch­marsh, Weyl, Wien­er, G. C. and W. H. Young. Per­haps it is even ap­pro­pri­ate to men­tion that Can­tor’s the­ory of transfin­ite num­bers has its ori­gin in a prob­lem in­volving tri­go­no­met­ric series. The present status and prom­in­ence of har­mon­ic ana­lys­is, however, is due in large part to Ant­oni Zyg­mund and the school that he cre­ated in the United States.

We shall first say a few words about Ant­oni Zyg­mund and try to ex­plain why he was able to es­tab­lish such a large and in­flu­en­tial school. By do­ing this, we shall also de­scribe, briefly, the field of har­mon­ic ana­lys­is and the vis­ion Zyg­mund had for this dis­cip­line. We then present a two-gen­er­a­tion “math­em­at­ic­al gene­a­logy” of Zyg­mund’s stu­dents and their stu­dents. We do this for two reas­ons. First, we be­lieve that this is the most con­crete evid­ence we can provide for gauging the in­flu­ence Zyg­mund had. Second, such a com­pil­a­tion may be a most use­ful doc­u­ment for a his­tor­i­an of math­em­at­ics.

Ant­oni Zyg­mund was born in Warsaw, Po­land, on Decem­ber 26, 1900. After com­plet­ing high school, he en­rolled in the Uni­versity of Warsaw in 1919. A few months later, he en­lis­ted in the Pol­ish army where he served dur­ing the cre­ation of the state of Po­land. He re­turned to Warsaw when the fight­ing ceased and gradu­ated from the Uni­versity in 1923. He stud­ied with Aleksander Ra­jch­man and de­voted him­self to the study of tri­go­no­met­ric series. He and Ra­jch­man wrote some joint pa­pers on sum­mab­il­ity the­ory. An­oth­er of his teach­ers was Wacław Si­er­piński with whom he pub­lished a pa­per in 1923. While still a stu­dent, he met Saks, who was three years older. Saks had a sig­ni­fic­ant in­flu­ence on Zyg­mund. They wrote some joint pa­pers and later pro­duced an ex­cel­lent text on the the­ory of func­tions.

He began his teach­ing ca­reer at the Warsaw Poly tech­nic­al School. From 1926 to 1930, he held the po­s­i­tion of “Privat Dozent” at the Uni­versity of Warsaw. Dur­ing these years in his nat­ive city, Zyg­mund’s math­em­at­ic­al activ­ity (mostly in the field of tri­go­no­met­ric series) was in­tense. He spent the aca­dem­ic year 1929–1930 in Eng­land as a Rock­e­feller Fel­low at the Uni­versit­ies of Ox­ford and Cam­bridge. There he met both Hardy and Lit­tle­wood as well as oth­ers who shared his sci­entif­ic in­terests. In par­tic­u­lar, it was there that the seeds of an im­port­ant col­lab­or­a­tion with R. E. A. C. Pa­ley were sown. He also met Norbert Wien­er with whom he and Pa­ley later wrote a sem­in­al pa­per that showed the im­port­ant re­la­tion­ship prob­ab­il­ity has with the the­ory of Four­i­er series. Dur­ing the ten months in Eng­land, he wrote ten pa­pers.

In the sum­mer of 1930, Zyg­mund was ap­poin­ted As­so­ci­ate Pro­fess­or of math­em­at­ics at the Uni­versity of Wilno. He stayed there un­til March 1940, when, to­geth­er with his wife and son, he man­aged to es­cape from oc­cu­pied Po­land. The ten-year peri­od in Wilno was a re­mark­ably pro­duct­ive one. His unique abil­ity to in­teg­rate ideas from many fields and his sense of dir­ec­tion on vari­ous sub­jects are evid­ent from his pub­lic­a­tions dur­ing this dec­ade. His col­lab­or­a­tion with Pa­ley poin­ted the way to the many con­nec­tions between the the­ory of func­tions and the study of Four­i­er series. With Pa­ley and Wien­er, he showed the im­port­ant ties between this last top­ic and prob­ab­il­ity the­ory. In Wilno, he dis­covered a bril­liant youth, Josef Mar­cinkiew­iez. It is one of the many tra­gedies of the second World War that this very tal­en­ted man died in the spring of 1940 when he was serving as an of­ficer in the Pol­ish army. To­geth­er with Mar­cinkiewicz, Zyg­mund ex­plored and pi­on­eered in oth­er fields of ana­lys­is. This ef­fort in­cluded an im­port­ant pa­per on the dif­fer­en­ti­ab­il­ity of mul­tiple in­teg­rals (an­oth­er young math­em­atician, Jessen, was in­volved in this re­search as well). Much of the sub­sequent study of func­tions of sev­er­al real vari­ables de­pends on the ideas in this work. Per­haps the most im­port­ant achieve­ment of this peri­od was the pub­lic­a­tion of the first edi­tion of his fam­ous book Tri­go­no­met­ric­al Series. In this book, one can find prac­tic­ally all the im­port­ant res­ults that were known on this sub­ject, as well as its con­nec­tions with oth­er dis­cip­lines. In ad­di­tion to the top­ics we have already men­tioned, the book in­cludes sub­jects and points of view that were new at that time. In par­tic­u­lar, one should keep in mind that it was dur­ing this peri­od that much of mod­ern func­tion­al ana­lys­is was de­veloped in Po­land by Banach and oth­ers. In Zyg­mund’s book, one can find the treat­ment of func­tion spaces and op­er­at­ors on them that is much in the spir­it of this new top­ic. It was in this work that the im­port­ance of the M. Riesz Con­vex­ity The­or­em, as a tool for study­ing op­er­at­ors, was made evid­ent.

Thanks to the ef­forts of J. D. Tamar­kin, Norbert Wien­er, and Jerzy Ney­man, in 1940 he re­ceived an of­fer of a vis­it­ing pro­fess­or­ship at M.I.T. as well as a visa to the United States. The Amer­ic­an aca­dem­ic world, at that time, was fa­cing many prob­lems. Zyg­mund had to start his Amer­ic­an ca­reer from the be­gin­ning. From 1940 to 1945, he was an as­sist­ant pro­fess­or at Mount Holy­oke Col­lege. Dur­ing this peri­od, he was also gran­ted a leave of ab­sence to spend the aca­dem­ic year 1942–1943 at the Uni­versity of Michigan. This, too, was a pro­lif­ic peri­od for Zyg­mund. He pro­duced el­ev­en pa­pers. His col­lab­or­a­tion with Raphael Salem began at this time. A little-known fact is that one of these pa­pers, with Tamar­kin, con­tains the el­eg­ant proof of the M. Riesz Con­vex­ity The­or­em that is known as the “Thor­in proof.” This proof gave birth to the “com­plex meth­od” in the the­ory of in­ter­pol­a­tion of op­er­at­ors. Thor­in did ob­tain his proof earli­er (in 1942), but he did not pub­lish it un­til 1947. Zyg­mund ac­know­ledged Thor­in’s pri­or­ity and al­ways re­ferred to the res­ult in­volved as the “Riesz–Thor­in The­or­em.” All this was done des­pite the very heavy teach­ing sched­ule (by mod­ern stand­ards) of nine hours per week. We should add that of­ten, dur­ing his ca­reer in Po­land, Zyg­mund had com­par­ably heavy teach­ing du­ties.

In 1945, Zyg­mund ac­cep­ted an as­so­ci­ate pro­fess­or­ship at the Uni­versity of Pennsylvania where he stayed un­til 1947. In that year, he was in­vited to join the fac­ulty at the Uni­versity of Chica­go where he spent the rest of his ca­reer. This was the be­gin­ning of an ex­cep­tion­al peri­od for Zyg­mund and, more gen­er­ally, for math­em­at­ics. Un­der the lead­er­ship of its chan­cel­lor, Robert M. Hutchins, the Uni­versity of Chica­go be­came a world lead­er in many aca­dem­ic fields. In par­tic­u­lar, Hutchins hired Mar­shall H. Stone who built an ex­cep­tion­al de­part­ment of math­em­at­ics in the en­su­ing years. In ad­di­tion to Zyg­mund, he brought many dis­tin­guished math­em­aticians to this de­part­ment. S. Mac Lane, S. S. Chern, and A. Weil were some of the seni­or men that joined well-known pro­fess­ors already in the de­part­ment: A. Ad­ri­an Al­bert, E. P. Lane, and L. M. Graves. The more ju­ni­or new­comers who came de­veloped in­to well-known lead­ers in their fields. I. Ka­plansky, P. Hal­mos, and I. E. Segal were some of these. Dis­tin­guished vis­it­ors from all over the world spent vari­ous peri­ods of time at the Uni­versity of Chica­go. J. E. Lit­tle­wood, M. Riesz, L. Hör­mander, S. Smale, and R. Salem rep­res­ent only a very small and ar­bit­rar­ily chosen sample of this group. In ad­di­tion to all this, a large num­ber of ex­traordin­ary gradu­ate stu­dents came to Chica­go to study with this il­lus­tri­ous group.

Zyg­mund flour­ished in this at­mo­sphere. Many of the tal­en­ted young people who came to study in Chica­go be­came his stu­dents. In ad­di­tion, he went to Ar­gen­tina in 1949 on a Ful­bright fel­low­ship where he dis­covered two out­stand­ing stu­dents, Al­berto Calder­ón and Mis­cha Cot­lar. Both went to Chica­go and soon earned their Ph.D.’s with him. Calder­ón soon be­came Zyg­mund’s col­lab­or­at­or, and their joint work is of such im­port­ance that many refer to the school we are dis­cuss­ing as the “Zyg­mund–Calder­ón school.” Though this name ap­pro­pri­ately clas­si­fies an im­port­ant por­tion of har­mon­ic ana­lys­is, it does not cov­er all that should be re­ferred to as the “Zyg­mund school.”

It is im­port­ant to real­ize the fol­low­ing unique fea­tures of this school. When Zyg­mund came to Chica­go, the “trend” in math­em­at­ics was very much in­flu­enced by the Bourbaki school and oth­er forces that cham­pioned a rather ab­stract and al­geb­ra­ic ap­proach for all of math­em­at­ics. Zyg­mund’s ap­proach to­ward his math­em­at­ics was very con­crete. He felt that it was most im­port­ant to ex­tend the more clas­sic­al res­ults in Four­i­er ana­lys­is to oth­er set­tings, to show the con­nec­tions of this field to oth­ers (as we have already in­dic­ated in this art­icle) and to dis­cov­er meth­ods for car­ry­ing this out. He real­ized that fun­da­ment­al ques­tions of cal­cu­lus and ana­lys­is were still not well un­der­stood. In a sense, he was “buck­ing the mod­ern trends.” In ret­ro­spect, his ap­proach proved to be very suc­cess­ful. This is seen not only by what we state here (his achieve­ments and the two-gen­er­a­tion gene­a­logy that in­cludes more than 170 names), but by the fact that the very con­crete prob­lems posed by Zyg­mund, with well-defined scope, at­trac­ted many of the very gif­ted stu­dents in Chica­go to work with him.

Zyg­mund con­tin­ued mak­ing im­port­ant con­tri­bu­tions. Per­haps the most sig­ni­fic­ant is the second edi­tion of his book Tri­go­no­met­ric­al Series. This two-volume work, pub­lished in 1959, in­cludes all that was in the earli­er edi­tion in ad­di­tion to most of the de­vel­op­ment in the field that oc­curred in the twenty-five years after the first edi­tion was writ­ten. This was a tre­mend­ous ef­fort for Zyg­mund. He com­plained to J. E. Lit­tle­wood that writ­ing this book cost him at least thirty re­search pa­pers. Lit­tle­wood replied that the book was worth more than twice that many good pa­pers. His work with Calder­ón, of course, was of para­mount im­port­ance. Even be­fore he met Calder­ón, he of­ten said that “the fu­ture of har­mon­ic ana­lys­is lies in sev­er­al di­men­sions.” The Calder­ón–Zyg­mund the­ory is a gi­ant step in this dir­ec­tion. They de­veloped a the­ory of “sin­gu­lar in­teg­ral op­er­at­ors” that has led to many ad­vances in the the­ory of par­tial dif­fer­en­tial equa­tions and many oth­er fields.

By 1956, Zyg­mund had trained the three stu­dents, Calder­ón, Eli­as M. Stein, and Guido Weiss, who were to form the back­bone of the Zyg­mund school, not only be­cause of their re­search con­tri­bu­tion, but be­cause of the large num­ber of stu­dents they have trained, a total of sev­enty-three to date (a num­ber that will prob­ably in­crease to sev­enty-sev­en by the time this art­icle is prin­ted). He con­tin­ued hav­ing stu­dents un­til 1971. Even after that date, however, he was act­ive math­em­at­ic­ally. Soon after com­ing to Chica­go, he or­gan­ized a weekly sem­in­ar that con­sisted of a one-hour present­a­tion of a cur­rent top­ic fol­lowed by an in­form­al hour of dis­cus­sion. This dis­cus­sion was open to any­one who wanted to present an idea or for­mu­late a prob­lem. This “Zyg­mund Sem­in­ar” con­tin­ued un­der his lead­er­ship through the sev­en­ties and early eighties.

We have de­scribed, briefly, some of Zyg­mund’s work, vis­ion, and in­flu­ence in the study of Four­i­er series and in­teg­rals. We in­dic­ated that he was a pi­on­eer in show­ing how this field was con­nec­ted with the the­ory of func­tions, prob­ab­il­ity the­ory, func­tion­al ana­lys­is, ana­lys­is in high­er-di­men­sion­al Eu­c­lidean spaces, and par­tial dif­fer­en­tial equa­tions. A more thor­ough bio­graphy would in­dic­ate an even broad­er vis­ion. He showed the im­port­ance of cer­tain func­tion spaces: \( L \log L \), the weak type spaces, the space of smooth func­tions (he was most proud of this cre­ation). He paved the way to oth­er top­ics in high­er di­men­sions by be­ing the first to es­tab­lish im­port­ant res­ults in the the­ory of Hardy spaces in­volving ana­lyt­ic func­tions of sev­er­al vari­ables. By writ­ing a beau­ti­ful pa­per on the Mar­cinkiewicz In­ter­pol­a­tion The­or­em (after Mar­cinkiewicz’s death), he led the way to “the real meth­od” in the the­ory of in­ter­pol­a­tion of op­er­at­ors. His col­lec­ted works have been com­piled and in­clude more than 150 pub­lic­a­tions. We give a pre­cise ref­er­ence to this volume at the end of this art­icle, where we cite some oth­er works con­tain­ing rel­ev­ant his­tor­ic­al ma­ter­i­al.

Zyg­mund’s per­son­al­ity con­trib­uted greatly to the in­flu­ence he had on his stu­dents and col­leagues. He was gentle, gen­er­ous, and friendly. His in­terests al­ways ex­ten­ded way bey­ond math­em­at­ics. Lit­er­at­ure and cur­rent events oc­cu­pied a con­sid­er­able amount of his at­ten­tion. The be­gin­ning of each day was de­voted to a thor­ough read­ing of the New York Times, and he ended the day en­grossed in a book; but math­em­at­ics was his pas­sion. His out­look on life and his con­sid­er­able sense of hu­mor al­most al­ways were con­nec­ted with math­em­at­ics. Once when walk­ing past a lounge in the Uni­versity of Chica­go that was filled with a loud crowd watch­ing TV, he asked one of his stu­dents what was go­ing on. The stu­dent told him that the crowd was watch­ing the World Series and ex­plained to him some of the fea­tures of this base­ball phe­nomen­on. Zyg­mund thought about it all for a few minutes and com­men­ted, “I think it should be called the World Se­quence.” On an­oth­er oc­ca­sion, after passing through sev­er­al rooms in a mu­seum filled with the paint­ings of a rather well-known mod­ern paint­er, he mused, “Math­em­at­ics and art are quite dif­fer­ent. We could not pub­lish so many pa­pers that used, re­peatedly, the same idea and still com­mand the re­spect of our col­leagues.” His judge­ments of oth­ers, however, was usu­ally kind. Once, when dis­cuss­ing the philo­sophy of writ­ing let­ters of re­com­mend­a­tion, he said to one of his stu­dents, “Con­cen­trate only on the achieve­ments, and ig­nore the mis­takes. When judging a math­em­atician you should only in­teg­rate \( \mathrm{f} + \) (the pos­it­ive part of his func­tion) and ig­nore the neg­at­ive part. Per­haps this should ap­ply more gen­er­ally to all eval­u­ations of your fel­low men.” Des­pite his con­sid­er­able achieve­ments, he al­ways con­sidered oth­ers as his equal and made his stu­dents feel at ease with him. He was al­ways easy to ap­proach and en­cour­aged stu­dents to come and talk with him. His of­fice was of­ten filled with stu­dents and col­leagues.

The genealogy

The fol­low­ing is a list of all of Zyg­mund’s Ph.D. stu­dents in the U.S. in chro­no­lo­gic­al or­der. Un­der each stu­dent, in­den­ted, is a list of all his or her stu­dents (through 1987), also in chro­no­lo­gic­al or­der. Each entry lists the cur­rent af­fil­i­ation if known, the date the Ph.D. was gran­ted, the uni­versity grant­ing the Ph.D., and the thes­is title. Zyg­mund also had four Ph.D. stu­dents in Po­land: L. Jas­manow­icz, Z. Lepecki, J. Mar­cinkiewicz, and K. Sokol-Soko­lowski; the last three are de­ceased.

Be­fore present­ing this list, let us make a few ob­ser­va­tions about such a gene­a­logy. Such a list has to be ter­min­ated some­where. We have chosen to lim­it ourselves to the second gen­er­a­tion since the in­flu­ence of Zyg­mund as a teach­er would be quite di­luted by the third gen­er­a­tion. We are aware that there are quite a few math­em­aticians who either totally or par­tially re­trained un­der Zyg­mund and his stu­dents, but do not show up on our list. One of us (Coi­f­man), for ex­ample, was a stu­dent of Kara­mata, but stud­ied in­tens­ively un­der Guido Weiss and, later, Calder­ón and Zyg­mund. We are also aware that a Ph.D. stu­dent may have more than one ad­visor. For ex­ample, when Calder­ón and Zyg­mund were at the Uni­versity of Chica­go to­geth­er, they had com­mon stu­dents. A con­sequence is that those of­fi­cially lis­ted as Zyg­mund stu­dents have their stu­dents on our list, while those lis­ted as Calder­ón stu­dents do not. A sim­il­ar situ­ation oc­curred at Wash­ing­ton Uni­versity between Coi­f­man and Weiss (the Coi­f­man stu­dents do not ap­pear on our list). To the best of our know­ledge, our list re­flects the ad­visor-stu­dent re­la­tion that was giv­en to us by the de­part­ments of math­em­at­ics in­volved. We know that there are many who have made sig­ni­fic­ant con­tri­bu­tions to the Zyg­mund school but who are not men­tioned here. We of­fer our apo­lo­gies to them for this and ask for their un­der­stand­ing.

The stu­dents of Zyg­mund are lis­ted in bold­face. The second gen­er­a­tion’s names are in­den­ted and are lis­ted be­low the name of their ad­visor.


We are grate­ful to Deena Ber­ton and Thomas Mahr, who par­ti­cip­ated in the early stages of this work. We are grate­ful to the many stu­dents of Zyg­mund who provided us with in­form­a­tion for the gene­a­logy and per­son­al re­mem­brances and who sug­ges­ted im­prove­ments in the text. In par­tic­u­lar, we want to thank Mis­cha Cot­lar, Eu­gene Fabes, Ben­jamin Mucken­houpt, Cora Sa­d­osky, Eli Stein, Daniel Wa­ter­man, Guido Weiss, and Richard Wheeden.

Zygmund’s Ph.D. Students in the U.S.

  • Nath­an J. Fine
    Re­tired, Pennsylvania State Uni­versity
    Ph.D. 1946, Uni­versity of Pennsylvania
    “On the Walsh Func­tions”

    • Justin J. Price
      Purdue Uni­versity
      Ph.D. 1956, Uni­versity of Pennsylvania
      “Some Ques­tions about Walsh Func­tions”

    • An­thony W. Hager
      Wes­ley­an Uni­versity
      Ph.D. 1965, Pennsylvania State Uni­versity
      “On the Tensor Product of Func­tion Rings”

    • Wil­li­am A. Webb
      Wash­ing­ton State Uni­versity
      Ph.D. 1969, Pennsylvania State Uni­versity
      “Auto­morph­isms of Form­al Puiseux Series”

  • Ching-Tsu Loo
    Ph.D. 1948, Uni­versity of Chica­go
    “Note on the Prop­er­ties of Four­i­er Coef­fi­cients”

  • Al­berto Calder­ón
    Buenos Aires, Ar­gen­tina
    Ph.D. 1950, Uni­versity of Chica­go
    I. “On the Er­god­ic The­or­em”
    II. “On the Be­ha­vi­or of Har­mon­ic Func­tions at the Bound­ary”
    III. “On the The­or­em of Mar­cinkiewicz and Zyg­mund”

    • Robert T. See­ley
      Uni­versity of Mas­sachu­setts, Bo­ston
      Ph.D. 1959, M.I.T.
      “Sin­gu­lar In­teg­rals on Com­pact Man­i­folds”

    • Ir­win S. Bern­stein
      City Col­lege, CUNY
      Ph.D. 1959, M.I.T.
      “On the Unique Con­tinu­ation Prob­lem of El­lipt­ic Par­tial Dif­fer­en­tial Equa­tions”

    • Is­rael Nor­man Katz
      Wash­ing­ton Uni­versity, Dept. of Sys­tems, Sci­ence and Math.,
      St. Louis, Mis­souri
      Ph.D. 1959, M.I.T.
      “On the Ex­ist­ence of Weak Solu­tions to Lin­ear Par­tial Dif­fer­en­tial Equa­tions”

    • Jerome H. Neuwirth
      Uni­versity of Con­necti­c­ut
      Ph.D. 1959, M.I.T.
      “Sin­gu­lar In­teg­rals and the Totally Hy­per­bol­ic Equa­tion”

    • Earl Berkson
      Uni­versity of Illinois
      Ph.D. 1961, Uni­versity of Chica­go
      I. “Gen­er­al­ized Di­ag­on­able Op­er­at­ors”
      II. “Some Met­rics on the Sub­spaces of a Banach Space”

    • Ev­elio To­mas Ok­lander
      Ph.D. 1964, Uni­versity of Chica­go
      “On In­ter­pol­a­tion of Banach Spaces”

    • Cora S. Sa­d­osky
      Howard Uni­versity
      Ph.D. 1965, Uni­versity of Chica­go
      “On Class Pre­ser­va­tion and Point­wise Con­ver­gence for Para­bol­ic Sin­gu­lar Op­er­at­ors”

    • Steph­en Vági
      De­Paul Uni­versity
      Ph.D. 1965, Uni­versity of Chica­go
      “On Mul­ti­pli­ers and Sin­gu­lar In­teg­rals in \( L_{\mkern-2mup} \) Spaces of Vec­tor Val­ued Func­tions”

    • Nestor Rivire
      Ph.D. 1966, Uni­versity of Chica­go
      “In­ter­pol­a­tion The­ory in \( S \)-Banach Spaces”

    • John C. Polk­ing
      Rice Uni­versity
      Ph.D. 1966, Uni­versity of Chica­go
      “Bound­ary Value Prob­lems for Para­bol­ic Sys­tems of Dif­fer­en­tial Equa­tions”

    • Um­berto Neri
      Uni­versity of Mary­land
      Ph.D. 1966, Uni­versity of Chica­go
      “Sin­gu­lar In­teg­ral Op­er­at­ors on Man­i­folds”

    • Miguel De Guzmán
      Uni­ver­sid­ad Com­plutense de Mad­rid
      Ph.D. 1967, Uni­versity of Chica­go
      “Sin­gu­lar In­teg­ral Op­er­at­ors with Gen­er­al­ized Ho­mo­gen­eity”

    • Car­los Segovia
      Uni­ver­sid­ad de Buenos Aires
      Ph.D. 1967, Uni­versity of Chica­go
      “On the Area Func­tion of Lus­in”

    • Keith Wil­li­am Powers
      Ph.D. 1972, Uni­versity of Chica­go
      “A Bound­ary Be­ha­vi­or Prob­lem in Pseudo-dif­fer­en­tial Op­er­at­ors”

    • Al­berto Torch­in­sky
      In­di­ana Uni­versity
      Ph.D. 1972, Uni­versity of Chica­go
      “Sin­gu­lar In­teg­rals in Lipschitz Spaces of Func­tions and Dis­tri­bu­tions”

    • Robert R. Re­it­ano
      Seni­or Fin­an­cial Of­ficer for John Han­cock
      Ph.D. 1976, M.I.T.
      “Bound­ary Val­ues and Re­stric­tions of Gen­er­al­ized Func­tions with Ap­plic­a­tions”

    • Josefina Dolores Al­varez Alonso
      Flor­ida At­lantic Uni­versity
      Ph.D. 1976, Uni­ver­sid­ad de Buenos Aires
      “Pseudo Dif­fer­en­tial Op­er­at­ors with Dis­tri­bu­tion Sym­bols”

    • Telma Cap­utti
      Uni­ver­sid­ad de Buenos Aires
      Ph.D. 1976, Uni­ver­sid­ad de Buenos Aires
      “Lipschitz Spaces”

    • Car­los Kenig
      Uni­versity of Chica­go
      Ph.D. 1978, Uni­versity of Chica­go
      \( H_{\mkern-2mup} \) Spaces on Lipschitz Do­mains”

    • An­gel Eduardo Gatto
      De­Paul Uni­versity
      Ph.D. 1979, Uni­ver­sid­ad de Buenos Aires
      “An Atom­ic De­com­pos­i­tion of Dis­tri­bu­tions in Para­bol­ic \( H_{\mkern-2mup} \) Spaces”

    • Cris­ti­an E. Gu­ti­er­rez
      Temple Uni­versity
      Ph.D. 1979, Uni­ver­sid­ad de Buenos Aires
      “Con­tinu­ity Prop­er­ties of Sin­gu­lar In­teg­ral Op­er­at­ors”

    • Kent Merry­field
      Cali­for­nia State Univ., Long Beach
      Ph.D. 1980, Uni­versity of Chica­go
      \( H_{\mkern-2mup} \) Spaces in Poly-Half Spaces”

    • F. Mi­chael Christ
      Ph.D. 1982, Uni­versity of Chica­go
      “Re­stric­tion of the Four­i­er Trans­form to Sub­man­i­folds of Low Codi­men­sion”

    • Ger­ald Co­hen
      Ph.D. 1982, Uni­versity of Chica­go
      “Hardy Spaces: Atom­ic De­com­pos­tion, Area Func­tions, and Some New Spaces of Dis­tri­bu­tions”

    • Maria Amelia Muschi­etti
      Na­tion­al Uni­versity of La Plata, Ar­gen­tina
      Ph.D. 1984, Na­tion­al Uni­versity of la Plata
      “On Com­plex Powers of El­lipt­ic Op­er­at­ors”

    • Marta Urciuolo
      Na­tion­al Uni­versity of Cor­doba, Ar­gen­tina
      Ph.D. 1985, Uni­versity of Buenos Aires
      “Sin­gu­lar In­teg­rals on Rec­ti­fi­able Sur­faces”

  • Beth­umne Vander­burg
    Ph.D. 1951, Uni­versity of Chica­go
    “Lin­ear Com­bin­a­tions of Haus­dorff Sum­mab­il­ity Meth­ods”

  • Henry Wil­li­am Oliv­er
    Pro­fess­or Emer­it­us Wil­li­ams Col­lege (Re­tired 1981)
    Ph.D. 1951, Uni­versity of Chica­go
    “Dif­fer­en­tial Prop­er­ties of Real Func­tions”

  • George Klein
    Ph.D. 1951, Uni­versity of Chica­go
    “On the Ap­prox­im­a­tion of Func­tions by Poly­no­mi­als”

  • Richard P. Gos­selin
    Uni­versity of Con­necti­c­ut
    Ph.D. 1951, Uni­versity of Chica­go
    “The The­ory of Loc­al­iz­a­tion for Double Tri­go­no­met­ric Series”

    • Richard Mont­gomery
      Uni­versity of Con­necti­c­ut, Gro­ton
      Ph.D. 1973, Uni­versity of Con­necti­c­ut
      “Closed Sub-al­gebra of Group Al­gebra”

  • Le­onard D. Berkovitz
    Purdue Uni­versity
    Ph.D. 1951, Uni­versity of Chica­go
    I. “Cir­cu­lar Sum­ma­tion and Loc­al­iz­a­tion of Double Tri­go­no­met­ric Series”
    II. “On Double Tri­go­no­met­ric In­teg­rals”
    III. “On Double Sturm–Li­ouville Ex­pan­sions”

    • Har­vey Thomas Banks
      Brown Uni­versity
      Ph.D. 1967, Purdue Uni­versity
      “Op­tim­al Con­trol Prob­lems with Delays”

    • Li­an Dav­id Sab­bagh
      Sab­bagh As­so­ci­ates, Inc.
      Ph.D. 1967, Purdue Uni­versity
      “Vari­ation­al Prob­lems with Lags”

    • Thomas Hack
      Ph.D. 1970, Purdue Uni­versity
      “Suf­fi­cient Con­di­tions in Op­tim­al Con­trol The­ory and Dif­fer­en­tial Games”

    • Jerry Searcy
      Ph.D. 1970, Purdue Uni­versity
      “Non­clas­sic­al Vari­ation­al Prob­lems Re­lated to an Op­tim­al Fil­ter Prob­lem”

    • Ral­ph Weath­er­wax
      Ph.D. 1972, Purdue Uni­versity
      “Lag­range Mul­ti­pli­ers for Ab­stract Op­tim­al Con­trol Pro­gram­ming Prob­lems”

    • Wil­li­am Brown­ing
      Ap­plied Math. Inc.
      Ph.D. 1974, Purdue Uni­versity
      “A Class of Vari­ation­al Prob­lems”

    • Gary R. Bates
      Murphy Oil
      Ph.D. 1977, Purdue Uni­versity
      “Hered­it­ary Op­tim­al Con­trol Prob­lems”

    • Ne­g­ash G. Med­him
      At­lanta Uni­versity
      Ph.D. 1980, Purdue Uni­versity
      “Ne­ces­sary con­di­tions for Op­tim­al Con­trol Prob­lems with Bounded State by a Pen­alty Meth­od”

    • Jiong­min Yong
      Uni­versity of Texas, Aus­tin
      Ph.D. 1986, Purdue Uni­versity
      “On Dif­fer­en­tial Games of Eva­sion and Pur­suit”

  • Vic­tor L. Sha­piro
    Uni­versity of Cali­for­nia at River­side
    Ph.D. 1952, Uni­versity of Chica­go
    “Square Sum­ma­tion and Loc­al­iz­a­tion of Double Tri­go­no­met­ric Series”
    “Sum­mab­il­ity of Double Tri­go­no­met­ric In­teg­rals”
    “Cir­cu­lar Sum­mab­il­ity \( C \) of Double Tri­go­no­met­ric Series”

    • Aaron Siegel
      Ph.D. 1958, Rut­gers Uni­versity
      “Sum­mab­il­ity \( C \) of Series of Sur­face Spher­ic­al Har­mon­ics”

    • Robert Fesq
      Kenyon Col­lege
      Ph.D. 1962, Uni­versity of Ore­gon
      “Green’s For­mula, Lin­ear Con­tinu­ity, and Haus­dorff Meas­ure”

    • Richard Crit­tenden
      Port­land State Uni­versity
      Ph.D. 1963, Uni­versity of Ore­gon
      “A The­or­em on the Unique­ness of \( (C_{11}) \) Sum­mab­il­ity of Walsh Series”

    • Lawrence Harp­er
      Uni­versity of Cali­for­nia at River­side
      Ph.D. 1965, Uni­versity of Ore­gon
      “Ca­pa­city of Sets and Har­mon­ic Ana­lys­is on the Group \( 2^{\omega} \)

    • Lawrence Kroll
      Ph.D. 1967, Uni­versity of Cali­for­nia at River­side
      “The Unique­ness of Hermite Series Un­der Pois­son–Abel Sum­mab­il­ity”

    • Robert Hughes
      Boise State Uni­versity
      Ph.D. 1968, Uni­versity of Cali­for­nia at River­side
      “Bound­ary Be­ha­vi­or of Ran­dom Val­ued Heat Poly­no­mi­al Ex­pan­sions”

    • Wil­li­am R. Wade
      Uni­versity of Ten­ness­ee
      Ph.D. 1968, Uni­versity of Cali­for­nia at River­side
      “Unique­ness The­ory of the Haar and Walsh Series”

    • Stan­ton P. Phil­lip
      Uni­versity of Cali­for­nia at Santa Cruz
      Ph.D. 1969, Uni­versity of Cali­for­nia at River­side
      “Hankel Trans­forms and Gen­er­al­ized Axi­ally Sym­met­ric Po­ten­tials”

    • James Diederich
      Uni­versity of Cali­for­nia at Dav­is
      Ph.D. 1970, Uni­versity of Cali­for­nia at River­side
      “Re­mov­able Sets for Point­wise Solu­tions of El­lipt­ic Par­tial Dif­fer­en­tial Equa­tions”

    • Gary Lipp­man
      Cali­for­nia State Uni­versity, Hay­ward
      Ph.D. 1970, Uni­versity of Cali­for­nia at River­side
      “Spher­ic­al sum­mab­il­ity of Con­jug­ate Mul­tiple Four­i­er Series and In­teg­rals at the Crit­ic­al In­dex”

    • Richard Escobedo
      Ph.D. 1971, Uni­versity of Cali­for­nia at River­side
      “Sin­gu­lar Spher­ic­al Har­mon­ic Ker­nels and Spher­ic­al Sum­mab­il­ity of Mul­tiple Tri­go­no­met­ric In­teg­rals and Series”

    • Joseph A. Re­u­ter
      Ph.D. 1973, Uni­versity of Cali­for­nia at River­side
      “Unique­ness of Laguerre Series Un­der Pois­son–Abel Sum­mab­il­ity”

    • John Ba­sing­er
      Lock­heed, Ontario, Cali­for­nia
      Ph.D. 1974, Uni­versity of Cali­for­nia at River­side
      “Tri­go­no­met­ric Ap­prox­im­a­tion, Fréchet Vari­ation, and the Double Hil­bert Trans­form”

    • Charles Burch
      Ph.D 1976, Uni­versity of Cali­for­nia at River­side
      “The Dini Con­di­tion and a Cer­tain Non­lin­ear El­lipt­ic Sys­tem of Par­tial Dif­fer­en­tial Equa­tions”

    • Lawrence D. Di­Fiore
      Ph.D. 1977, Uni­versity of Cali­for­nia at River­side
      “Isol­ated Sin­gu­lar­it­ies and Reg­u­lar­ity of Cer­tain Non­lin­ear Equa­tions”

    • Dav­id Holmes
      TRW, San Bern­ardino, Cali­for­nia
      Ph.D. 1981, Uni­versity of Cali­for­nia at River­side
      “An Ex­ten­sion to \( n \)-di­men­sions of Cer­tain Non­lin­ear Equa­tions”

    • John C. Fay
      Cali­for­nia State Uni­versity, San Bern­ardino
      Ph.D. 1986, Uni­versity of Cali­for­nia at River­side
      “Second and High­er Or­der Quasi­lin­ear El­lipt­i­city on the \( N \)-tor­us”

  • Mis­cha Cot­lar
    Uni­ver­sid­ad Cent­ral de Venezuela
    Ph.D. 1953, Uni­versity of Chica­go
    “On the The­ory of Hil­bert Trans­forms”

    • Ra­fael Pan­zone
      Uni­ver­sid­ad Nacion­al del Sur, Bahia Blanca, Ar­gen­tina
      Ph.D. 1958, Uni­versity of Buenos Aires
      “On a Gen­er­al­iz­a­tion of Po­ten­tial Op­er­at­ors of the Riemann–Li­ouville Type”

    • Cora Ratto de Sa­d­osky
      De­ceased (1980)
      Ph.D. 1959, Uni­versity of Buenos Aires
      “Con­di­tions of Con­tinu­ity of Gen­er­al­ized Po­ten­tial Op­er­at­ors with Hy­per­bol­ic Met­ric”

    • Eduardo Ort­iz
      Im­per­i­al Col­lege, Lon­don
      Ph.D. 1961, Uni­versity of Buenos Aires
      “Con­tinu­ity of Po­ten­tial Op­er­at­ors in Spaces with Weighted Meas­ures”

    • Rodrigo Aro­cena
      Math­em­at­ics In­sti­tute, Mon­tevid­eo, Ur­uguay
      Ph.D. 1979, Uni­ver­sid­ad Cent­ral de Venezuela

  • George W. Mor­genthaler
    Uni­versity of Col­or­ado
    Ph.D. 1953, Uni­versity of Chica­go
    I. “The Cent­ral Lim­it The­or­em for Or­thonor­mal Sys­tems”
    II. “The Walsh Func­tions”

  • Daniel Wa­ter­man
    Syra­cuse Uni­versity
    Ph.D. 1954, Uni­versity of Chica­go
    I. “In­teg­rals As­so­ci­ated with Func­tions of \( L_p \)
    II. “A Con­ver­gence The­or­em”
    III. “On Some High In­di­cies The­or­ems”

    • Syed A. Hu­sain
      Ph.D. 1959, Purdue Uni­versity
      “Con­ver­gence Factors and Sum­mab­il­ity of Or­thonor­mal Ex­pan­sions”

    • Dan J. Eu­stice
      Ohio State Uni­versity
      Ph.D. 1960, Purdue Uni­versity
      “Sum­mab­il­ity of Or­tho­gon­al Series”

    • Don­ald W. So­lomon
      Uni­versity of Wis­con­sin, Mil­wau­kee
      Ph.D. 1966, Wayne State Uni­versity
      “Den­joy In­teg­ra­tion in Ab­stract Spaces”

    • Jo­gin­dar S. Ratti
      Ph.D. 1966, Wayne State Uni­versity
      “Gen­er­al­ized Riesz Sum­mab­il­ity”

    • George Gasper, Jr.
      North­west­ern Uni­versity
      Ph.D. 1967, Wayne State Uni­versity
      “On the Lit­tle­wood–Pa­ley and Lus­in Func­tions in High­er Di­men­sions”

    • James R. McLaugh­lin
      Ph.D.1968, Wayne State Uni­versity
      “On the Haar and Oth­er Clas­sic­al Or­thonor­mal Sys­tems”

    • Cor­nel­is W. On­neweer
      Uni­versity of New Mex­ico, Al­buquerque, NM
      Ph.D. 1969, Wayne State Uni­versity
      “On the Con­ver­gence of Four­i­er Series Over Cer­tain Zero-Di­men­sion­al Groups”

    • San­ford J. Per­l­man
      Ph.D. 1972, Wayne State Uni­versity
      “On the The­or­em of Fatou and Stepan­off”

    • Elaine Co­hen
      Uni­versity of Utah
      Ph.D. 1974, Syra­cuse Uni­versity
      “On the De­gree of Ap­prox­im­a­tion of a Func­tion by Par­tial Sums of its Four­i­er Series”

    • Dav­id Engles
      Ph.D. 1974, Syra­cuse Uni­versity
      “Bounded Vari­ation and its Gen­er­al­iz­a­tions”

    • Ar­thur D. Shind­helm
      Ph.D. 1974, Syra­cuse Uni­versity
      “Gen­er­al­iz­a­tions of the Banach—Saks Prop­erty”

    • Mi­chael J. Schramm
      LeMoyne Col­lege, Syra­cuse, N Y
      Ph.D. 1982, Syra­cuse Uni­versity
      “Top­ics in Gen­er­al­ized Bounded Vari­ation”

    • Pedro Isaza
      Ph.D. 1986, Syra­cuse Uni­versity
      “Func­tions of Gen­er­al­ized Bounded Vari­ation and Four­i­er Series”

    • Lawrence D’Ant­o­nio, Jr.
      SUNY at New Paltz
      Ph.D. 1986, Syra­cuse Uni­versity
      “Func­tions of Gen­er­al­ized Bounded Vari­ation. Sum­mab­il­ity of Four­i­er Series”

  • Iz­aak Wirszup
    Uni­versity of Chica­go
    Ph.D. 1955, Uni­versity of Chica­go
    “On an Ex­ten­sion of the Cesàro Meth­od of Sum­mab­il­ity to the Log­ar­ithmic Scale”

  • Eli­as M. Stein
    Prin­ceton Uni­versity
    Ph.D. 1955, Uni­versity of Chica­go
    “Lin­ear Op­er­at­ors on \( L_{\mkern-2mup} \) Spaces”

    • Steph­en Wainger
      Uni­versity of Wis­con­sin, Madis­on
      Ph.D. 1962, Uni­versity of Chica­go
      “Spe­cial Tri­go­no­met­ric­al Series in \( K \)-Di­men­sions”

    • Mitchell Her­bert Taibleson
      Wash­ing­ton Uni­versity in St. Louis
      Ph.D. 1963, Uni­versity of Chica­go
      “Smooth­ness and Dif­fer­en­ti­ab­il­ity Con­di­tions for Func­tions and Dis­tri­bu­tions on \( E_n \)

    • Robert S. Strichartz
      Cor­nell Uni­versity
      Ph.D. 1966, Prin­ceton Uni­versity
      “Mul­ti­pli­ers on Gen­er­al­ized So­bolev Spaces”

    • Nor­man J. Weiss
      Queens Col­lege, CUNY
      Ph.D. 1966, Prin­ceton Uni­versity
      “Al­most Every­where Con­ver­gence of Pois­son In­teg­rals on Tube Do­mains Over Cones”

    • Daniel A. Lev­ine
      Ph.D. 1968, Prin­ceton Uni­versity
      “Sin­gu­lar In­teg­ral Op­er­at­ors on Spheres”

    • Charles Louis Fef­fer­man
      Prin­ceton Uni­versity
      Ph.D. 1969, Prin­ceton Uni­versity
      “In­equal­it­ies for Strongly Sin­gu­lar Con­vo­lu­tion Op­er­at­ors”

    • Steph­en Samuel Gel­bart
      Weiz­mann In­sti­tute of Sci­ence, Is­rael
      Ph.D. 1970, Prin­ceton Uni­versity
      “Four­i­er Ana­lys­is on Mat­rix Space”

    • Lawrence Dick­son
      Ph.D. 1971, Prin­ceton Uni­versity
      “Some Lim­it Prop­er­ties of Pois­son In­teg­rals and Holo­morph­ic Func­tions on Tube Do­mains”

    • Steven G. Krantz
      Wash­ing­ton Uni­versity in St. Louis
      Ph.D. 1974, Prin­ceton Uni­versity
      “Op­tim­al Lipschitz and \( L_{\mkern-2mup} \) Es­tim­ates for the Equa­tion \( \bar{\partial}u = F \) on Strongly Pseudo-Con­vex Do­mains”

    • Wil­li­am Beck­ner
      Uni­versity of Texas, Aus­tin
      Ph.D. 1975, Prin­ceton Uni­versity
      “In­equal­it­ies in Four­i­er Ana­lys­is”

    • Robert A. Fef­fer­man
      Uni­versity of Chica­go
      Ph.D. 1975, Prin­ceton Uni­versity
      “A The­ory of En­tropy in Four­i­er Ana­lys­is”

    • Is­rael Zib­man
      Ph.D. 1976, Prin­ceton Uni­versity
      “Some Char­ac­ter­ist­ics of the \( n \)-Di­men­sion­al Peano De­riv­at­ive”

    • Gregg Jay Zuck­er­man
      Yale Uni­versity
      Ph.D. 1975, Prin­ceton Uni­versity
      “Some Char­ac­ter Iden­tit­ies for Semisimple Lie Groups”

    • Daryl Neil Geller
      SUNY at Stony Brook
      Ph.D. 1977, Prin­ceton Uni­versity
      “Four­i­er Ana­lys­is on the Heis­en­berg Group”

    • Duong Hong Phong
      Columbia Uni­versity
      Ph.D. 1977, Prin­ceton Uni­versity
      “On Hölder and \( L_{\mkern-2mup} \) Es­tim­ates for the \( \bar{\partial} \) Equa­tion on Strongly Pseudo-Con­vex Do­mains”

    • Dav­id Marc Gold­berg
      Sun Mi­crosys­tems, Pa­lo Alto, CA
      Ph.D. 1978, Prin­ceton Uni­versity
      “A Loc­al Ver­sion of Real Hardy Spaces”

    • Juan Car­los Per­al
      Fac­ultad de Cien­cias, Bil­bao, Spain
      Ph.D. 1978, Prin­ceton Uni­versity
      \( L_{\mkern-2mup} \) Es­tim­ates for the Wave Equa­tion”

    • Meir Shin­nar
      Ph.D. 1978, Prin­ceton Uni­versity
      “Ana­lyt­ic Con­tinu­ation of Group Rep­res­ent­a­tions”

    • Robert Mi­chael Beals
      Rut­gers Uni­versity
      Ph.D. 1980, Prin­ceton Uni­versity
      \( L_{\mkern-2mup} \) Bounded­ness of Cer­tain Four­i­er In­teg­ral Op­er­at­ors”

    • Dav­id Saul Jer­is­on
      Ph.D. 1980, Prin­ceton Uni­versity
      “The Di­rich­let Prob­lem for the Kohn Lapla­cian on the Heis­en­berg Group”

    • Charles Robin Gra­ham
      Uni­versity of Wash­ing­ton
      Ph.D. 1981, Prin­ceton Uni­versity
      “The Di­rich­let Prob­lem for the Berg­man Lapla­cian”

    • Al­lan T. Green­leaf
      Uni­versity of Rochester
      Ph.D. 1982, Priniceton Uni­versity
      “Prini­cip­al Curvature and Har­mon­ic Ana­lys­is”

    • An­drew Gran­ville Ben­nett
      Kan­sas State Uni­versity
      Ph.D. 1985, Prin­ceton Uni­versity
      “Prob­ab­il­ist­ic Square Func­tions, Mar­tin­gale Trans­forms and A Pri­ori Es­tim­ates”

    • Chris­toph­er Sogge
      Uni­versity of Chica­go
      Ph.D. 1985, Prin­ceton Uni­versity
      “Os­cil­lat­ory In­teg­rals and Spher­ic­al Har­mon­ics”

    • Robert Gross­man
      Uni­versity of Cali­for­nia, Berke­ley
      Ph.D. 1985, Prin­ceton Uni­versity
      “Small Time Loc­al Con­trol­lab­il­ity”

    • Kath­er­ine P. Diaz
      Texas A & M Uni­versity
      Ph.D. 1986, Prin­ceton Uni­versity
      “The Szegö \( K \) Ker­nel as a Sin­gu­lar In­teg­ral Ker­nel on a Weakly Pseudo-Con­vex Do­main”

    • Peter N. Heller
      Ph.D. 1986, Prin­ceton Uni­versity
      “Ana­lyti­city and Reg­u­lar­ity for Non­homo­gen­eous Op­er­at­ors on the Heis­en­berg Group”

    • C. An­drew Neff
      IBM, Wat­son Re­search Cen­ter, York­town Heights, N Y
      Ph.D. 1986, Prin­ceton Uni­versity
      “Max­im­al Func­tion Es­tim­ates for Mero­morph­ic Nevan­linna Func­tions”

    • Der-Chen Chang
      Uni­versity of Mary­land
      Ph.D. 1987, Prin­ceton Uni­versity
      “On \( L_{\mkern-2mup} \) and Hold­er Es­tim­ates for the \( \bar{\partial} \)-Neu­mann Prob­lem on Strongly Pseudo­con­vex Do­mains”

    • Sundaram Thangavelu
      Tata In­sti­tute, Ban­galore, In­dia
      Ph.D. 1987, Prin­ceton Uni­versity
      “Riesz Means and Mul­ti­pli­ers for Hermite Ex­pan­sions”

    • Hart F. Smith
      Mas­sachu­setts In­sti­tute of Tech­no­logy
      Ph.D. 1988, Prin­ceton Uni­versity
      “The Subel­lipt­ic Ob­lique De­riv­at­ive Prob­lem”

  • Wil­li­am J. Ri­ordan
    Ph.D. 1955, Uni­versity of Chica­go
    “On the In­ter­pol­a­tion of Op­er­a­tions”

  • Vivi­enne E. Mor­ley
    Ph.D. 1956, Uni­versity of Chica­go
    “Sin­gu­lar In­teg­rals”

  • Guido Leo­pold Weiss
    Wash­ing­ton Uni­versity in St. Louis
    Ph.D. 1956, Uni­versity of Chica­go
    “On Cer­tain Classes of Func­tion Spaces and on the In­ter­pol­a­tion of Sub­lin­ear Op­er­at­ors”

    • Jim­mie Ray Hatte­mer
      South­ern Illinois Uni­versity, Ed­wards­ville
      Ph.D. 1964, Wash­ing­ton Uni­versity
      “On Bound­ary Be­ha­vi­or of Tem­per­at­ures in Sev­er­al Vari­ables”

    • Richard Hunt
      Purdue Uni­versity
      Ph.D. 1965, Wash­ing­ton Uni­versity
      “Op­er­at­ors Act­ing on Lorentz Spaces”

    • Robert Og­den
      South­w­est Texas State Uni­versity
      Ph.D. 1970, Wash­ing­ton Uni­versity
      “Har­mon­ic Ana­lys­is on the Cone As­so­ci­ated with Non­com­pact Or­tho­gon­al Groups”

    • Robert Wil­li­am Latzer
      Ph.D. 1971, Wash­ing­ton Uni­versity
      “Non-Dir­ec­ted Light Sig­nals and the Struc­ture of Time”

    • Richard Ru­bin
      Flor­ida In­ter­na­tion­al Uni­versity
      Ph.D. 1974, Wash­ing­ton Uni­versity
      “Har­mon­ic Ana­lys­is on the Group of Ri­gid Mo­tions of the Eu­c­lidean Plane”

    • Roberto Ma­cias
      PEMA, Sante Fe, Ar­gen­tina
      Ph.D. 1974, Wash­ing­ton Uni­versity
      “In­ter­pol­a­tion The­or­ems on Gen­er­al­ized Hardy Spaces”

    • Roberto Gan­dulfo
      Uni­ver­sid­ade de Brasília, Brasil
      Ph.D. 1975, Wash­ing­ton Uni­versity
      “Mul­ti­pli­er Op­er­at­ors for Ex­pan­sions in Spher­ic­al Har­mon­ics and Ul­tra­s­pher­ic­al Poly­no­mi­als”

    • Minna Chao
      Ph.D. 1976, Wash­ing­ton Uni­versity
      “Har­mon­ic Ana­lys­is of a Second Or­der Sin­gu­lar Dif­fer­en­tial Op­er­at­or As­so­ci­ated with Non-Com­pact Semi-Simple Rank-One Lie Groups”

    • Mi­chael Hemler
      The Fuqua School of Busi­ness, Duke Uni­versity
      Ph.D. 1980, Wash­ing­ton Uni­versity
      “The Mo­lecu­lar The­ory of \( H^{p,q,s} (H^n) \)

    • José Dor­ron­soro
      Uni­ver­sid­ad Autónoma de Mad­rid
      Ph.D. 1981, Wash­ing­ton Uni­versity
      “Weighted Hardy Spaces on Her­mitian Hy­per­bol­ic Spaces”

    • Eu­genio Hernan­dez
      Uni­ver­sid­ad Autónoma de Mad­rid
      Ph.D. 1981, Wash­ing­ton Uni­versity
      “Top­ics in Com­plex In­ter­pol­a­tion”

    • Le­onardo Colzani
      Uni­versita de­gli Studi di Mil­ano
      Ph.D. 1982, Wash­ing­ton Uni­versity
      “Hardy and Lipschitz Spaces on Unit Spheres”

    • Fernando Sor­ia
      Uni­ver­sid­ad Autónoma de Mad­rid
      Ph.D. 1983, Wash­ing­ton Uni­versity
      “Classes of Func­tions Gen­er­ated by Blocks and As­so­ci­ated Hardy Spaces”

    • Han Yong Shen
      Pek­ing Uni­versity; presently on leave at Wash­ing­ton Uni­versity,
      Ph.D. 1984, Wash­ing­ton Uni­versity
      “Cer­tain Hardy-Type Spaces that can be Char­ac­ter­ized by Max­im­al Func­tions and Vari­ations of the Square Func­tions”

    • An­ita Tabacco Vig­nati
      Po­litec­nico di Torino, Torino, Italy
      Ph.D. 1986, Wash­ing­ton Uni­versity
      “In­ter­pol­a­tion of Quasi-Banach Spaces”

    • Marco Vig­nati
      Po­litec­nico di Torino, Torino, Italy
      Ph.D. 1986, Wash­ing­ton Uni­versity
      “In­ter­pol­a­tion: Geo­metry and Spec­tra”

    • Ales Za­loznik
      Uni­versity of Ljubljana, Yugoslavia
      Ph.D. 1987, Wash­ing­ton Uni­versity
      “Func­tion Spaces Gen­er­ated by Blocks As­so­ci­ated with Spheres, Lie Groups and Spaces of Ho­mo­gen­eous Type”

  • Mary Bish­op Weiss
    Ph.D. 1957, Uni­versity of Chica­go
    “The Law of the It­er­ated Log­ar­ithm for La­cun­ary Series and Ap­plic­a­tions to Hardy–Lit­tle­wood Series”

  • Paul Joseph Co­hen
    Stan­ford Uni­versity
    Ph.D. 1958, Uni­versity of Chica­go
    “Top­ics in the The­ory of Unique­ness of Tri­go­no­met­ric Series”

    • Peter Sarnak
      Stan­ford Uni­versity
      Ph.D. 1980, Stan­ford Uni­versity
      “Prime Geodes­ic The­or­ems”

  • Ben­jamin Mucken­houpt
    Rut­gers Uni­versity
    Ph.D. 1958, Uni­versity of Chica­go
    “On Cer­tain Sin­gu­lar In­teg­rals”

    • Eileen L. Poi­ani
      Saint Peter’s Col­lege, Jer­sey City, NJ
      Ph.D. 1971, Rut­gers Uni­versity
      “Mean Cesàro Sum­mab­il­ity of Laguerre and Hermite Series and Asymp­tot­ic Es­tim­ates of Laguerre and Hermite Poly­no­mi­als”

    • Hsiao-Wei Kuo
      Ph.D. 1975, Rut­gers Uni­versity
      “Mean Con­ver­gence of Jac­obi Series”

    • Ernst Adams
      Ph.D. 1981, Rut­gers Uni­versity
      “On Weighted Norm In­equal­it­ies for the Riesz Trans­forms of Func­tions with Van­ish­ing Mo­ments”

  • Efr­em Her­bert Os­trow
    Cali­for­nia State Uni­versity, North­ridge
    Ph.D. 1960, Uni­versity of Chica­go
    “A The­ory of Gen­er­al­ized Hil­bert Trans­forms”

  • Richard O’Neil
    SUNY at Al­bany
    Ph.D. 1960, Uni­versity of Chica­go
    “Frac­tion­al In­teg­ra­tion and Or­licz Spaces”

    • Jack Bry­ant
      Texas A & M Uni­versity
      Ph.D. Rice Uni­versity

    • Ger­aldo S. de Souza
      Au­burn Uni­versity
      Ph.D. 1980, SUNY at Al­bany
      “Spaces Formed by Spe­cial Atoms”

  • Mar­vin Barsky
    Beaver Col­lege, Glen­side, PA
    Ph.D. 1964, Uni­versity of Chica­go
    “On Re­peated Con­ver­gence of Series”

  • Chao Ping Chang
    Re­tired - Uni­versity of Auck­land, New Zea­l­and
    Ph.D. 1964, Uni­versity of Chica­go
    “On Cer­tain Ex­po­nen­tial Sums Arising in Con­jug­ate Mul­tiple Four­i­er Series”

  • Eu­gene Barry Fabes
    Uni­versity of Min­nesota
    Ph.D. 1965, Uni­versity of Chica­go
    “Para­bol­ic Par­tial Dif­fer­en­tial Equa­tions and Sin­gu­lar In­teg­rals”

    • Max Jodeit
      Uni­versity of Min­nesota
      Ph.D. 1967, Rice Uni­versity
      “Sym­bols of Para­bol­ic Sin­gu­lar In­teg­rals and Some \( L_{\mkern-2mup} \) Bound­ary Value Prob­lems”

    • Ju­lio Bouil­let
      In­sti­tuto Ar­gen­tino de Matem­at­ica, Buenos Aires, Ar­gen­tina
      Ph.D. 1972, Uni­versity of Min­nesota
      “Di­rich­let Prob­lem for Para­bol­ic Equa­tions with Con­tinu­ous Coef­fi­cients”

    • Steph­en Sroka
      De­part­ment of De­fense, Fort Meade, MD
      Ph.D. 1975, Uni­versity of Min­nesota
      “The Ini­tial-Di­rich­let Prob­lem for Para­bol­ic Par­tial Dif­fer­en­tial Equa­tions with Uni­formly Con­tinu­ous Coef­fi­cients and Data in \( L_{\mkern-2mup} \)

    • An­gel Gu­ti­er­rez
      Uni­ver­sid­ad Autónoma de Mad­rid, Mad­rid, Spain
      Ph.D. 1979, Uni­versity of Min­nesota
      “A Pri­ori \( L_{\mkern-2mup} \)-Es­tim­ates for the Solu­tion of the Navi­er Equa­tions of Elasti­city, Giv­en the Forles on the Bound­ary”

    • Gregory Ver­chota
      Uni­versity of Illinois at Chica­go
      Ph.D. 1982, Uni­versity of Min­nesota
      “Lay­er Po­ten­tials and Bound­ary Value Prob­lems for Laplace’s Equa­tion on Lipschitz Do­mains”

    • Pa­tri­cia Bau­man
      Purdue Uni­versity
      Ph.D. 1982, Uni­versity of Min­nesota
      “Prop­er­ties of Non-Neg­at­ive Solu­tions of Second Or­der El­lipt­ic Equa­tions and Their

    • Rus­sell Brown
      Uni­versity of Chica­go
      Ph.D. 1987, Uni­versity of Min­nesota
      “Lay­er Po­ten­tials and Bound­ary Value Prob­lems for the Heat Equa­tion in Lipschitz Do­mains”

  • Richard Lee Wheeden
    Rut­gers Uni­versity
    Ph.D. 1965, Uni­versity of Chica­go
    “On Tri­go­no­me­tirc Series As­so­ci­ated with Hy­per­sin­gu­lar In­teg­rals”

    • Ed­ward P. Lotkowski
      Ph.D. 1975, Rut­gers Uni­versity
      “Lipschitz Spaces with Weights”

    • Rus­sell T. John
      Ph.D. 1975, Rut­gers Uni­versity
      “Weighted Norm In­equal­it­ies for Sin­gu­lar and Hy­per­sin­gu­lar In­teg­rals”

    • Douglas S. Kur­tz
      New Mex­ico State Uni­versity
      Ph.D. 1978, Rut­gers Uni­versity
      “Lit­tle­wood–Pa­ley and Mulit­pli­er The­or­ems on Weighted \( L_{\mkern-2mup} \) Spaces”

  • J. Mar­shall Ash
    De­Paul Uni­versity
    Ph.D. 1966, Uni­versity of Chica­go
    “Gen­er­al­iz­a­tions of the Riemann De­riv­at­ive”

    • P. J. O’Con­nor
      Ph.D. 1969, Wes­ley­an Uni­versity
      “Gen­er­al­ized Dif­fer­en­ti­ation of Func­tions of a Real Vari­able”

  • I. Louis Gor­don
    Re­tired, Uni­versity of Illinois, Chica­go
    Ph.D. 1967, Uni­versity of Chica­go
    “Per­ron’s In­teg­ral for De­riv­at­ives in \( L_r \)

  • Yorham Sa­gh­er
    Uni­versity of Illinois at Chica­go
    Ph.D. 1967, Uni­versity of Chica­go
    “On Hy­per­sin­gu­lar In­teg­rals with Com­pez Ho­mo­gen­eity”

    • Mi­chael Cwikel
      Is­rael In­sti­tute of Tech­no­logy

    Sim Lasher
    Uni­versity of Illinois at Chica­go
    Ph.D. 1967, Uni­versity of Chica­go
    “On Dif­fer­en­ti­ation and De­riv­at­ives in \( L^r \)

  • Leo Frank Ziomek
    Ph.D. 1967, Uni­versity of Chica­go
    “On the Bound­ary Be­ha­vi­or in the Met­ric \( L_{\mkern-2mup} \) of Subhar­mon­ic Func­tions”

  • Wil­li­am C. Con­nett
    Uni­versity of Mis­souri at St. Louis
    Ph.D. 1969, Uni­versity of Chica­go
    “Form­al Mul­ti­plic­a­tion of Tri­go­no­met­ric Series and the No­tion of Gen­er­al­ized Con­jugacy”

  • Thomas Walsh
    Uni­versity of Flor­ida
    Ph.D. 1969, Uni­versity of Chica­go
    “Sin­gu­lar In­teg­rals of \( L^{\prime} \) func­tions”

  • Mar­vin J. Kohn
    Brook­lyn Col­lege, CUNY
    Ph.D. 1970, Uni­versity of Chica­go
    “Riemann Sum­mab­il­ity of Mul­tiple Tri­go­no­met­ric Series”

  • Styl­lanus C. Pi­chor­ides
    Uni­versity of Crete
    Ph.D. 1971, Uni­versity of Chica­go
    “On the Best Val­ues of the Con­stants in the The­or­ies of M. Riesz, Zyg­mund, and Kolmogorov”