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”
