#### 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.

#### Acknowledgments

#### 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

Deceased

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

Deceased

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

UCLA

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

Deceased

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

M.I.T.

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**

Deceased

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 UniversityGeraldo 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

Adjoints”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**

Deceased

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”