# Antoni Szczepan Zygmund

### Quick Info

Born
25 December 1900
Warsaw, Russian Empire (now Poland)
Died
30 May 1992
Chicago, Illinois, USA

Summary
Zygmund's work in harmonic analysis has application in the theory of waves and vibrations. He also did major work in Fourier analysis and its application to partial differential equations.

### Biography

Antoni Zygmund's father was Wincenty Zygmund and his mother was Antonina Perowska; both were of peasant origin. He was born into a partitioned Poland, the third of the three partitions took place in 1795 and split Poland between Russia, Prussia and Austria. Antoni was the eldest of his parents' four children with three younger sisters Jadwiga, Felicja and Maria. He attended elementary school in Warsaw, completing this stage of his education in 1912. He then entered the high school Seventh Gymnasium at 53 Zlota Street, Warsaw. After two years of secondary education World War I broke out in 1914 and Antoni, together with the rest of the Zygmund family, were evacuated to Poltava in the Ukraine. Antoni continued his education in Poltava then, in 1918, he returned to Poland which had become an independent country for the first time in well over 100 years. He attended the Kazimierz Kulwiee's Gymnasium for a year; it was a school specially set up to educate re-immigrants from Russia. In 1919 was awarded his school certificate and entered University of Warsaw:-
Since his boyhood Antoni Zygmund had been interested in astronomy and had a considerable knowledge in this field; at that time, however, such studies were not offered in Warsaw. He therefore decided to study mathematics.
At the University of Warsaw Zygmund was taught by, and very much influenced by, Janiszewski, Mazurkiewicz and Sierpiński. He attended lectures by Sierpiński on set theory, Mazurkiewicz on analytic functions, Samuel Dickstein on algebra and history of mathematics, and Stefan Kwietniewski on projective geometry. Zygmund, however, was most influenced by Aleksander Rajchman and by Stanisław Saks who had been his school friend, although three years older than he was. Aleksander Rajchman was interested in the theory of trigonometric series. He held lectures and a seminar on the topic for a few selected students. This gave Zygmund a life-long interest in the trigonometric series.

There was a forced break in Zygmund's studies in 1920 when he was called up for the army. He did not see active service in Polish-Russian war, however, and was able to return to his studies. In 1922 he was appointed (as was his friend Saks) to be an instructor in the Department of Mathematics at Warsaw Polytechnic School. Zygmund obtained his Ph.D. from the University of Warsaw in 1923 for a dissertation on the Riemannian theory of trigonometric series written under Aleksander Rajchman's supervision. His formal supervisor, however, was Mazurkiewicz since Rajchman was too junior to officially undertake the supervision role.

From 1922 to 1929 he taught at the Polytechnic School of Warsaw, being promoted from junior to senior instructor after obtaining his doctorate. On 12 February 1925 Zygmund married Irena Parnowska. They met when she was a mathematics student and, after taking her degree, she became a mathematics teacher. In 1926 Zygmund submitted his habilitation dissertation to the University of Warsaw and began teaching there. He continued to hold his position at the Polytechnic School as well.

He spent the year 1929-30 in England supported by a Rockefeller fellowship. The first half of this year he spent at Oxford with Hardy, then in the second half he studied at Cambridge with Littlewood. He became friend and collaborator with Paley at this time. In 1930 he was appointed to the chair of mathematics at the University of Stefan Batory in Vilnius, Poland (later Lithuania as we describe below). There he met Jozef Marcinkiewicz who was a first year student. Despite the difference in status between the two, they collaborated until the outbreak of World War II. Irena Zygmund worked in Vilnius as a school teacher until the birth of their first son Jerzy in 1935.

In going to Vilnius, Zygmund was going into a disputed city. The dispute centred on whether the city of Vilnius was part of Poland or of Lithuania. After a period under Polish control in 1919-20 the Red Army took the city in 1920 and handed control to Lithuania. In 1922, however, international agreement saw Vilnius become part of Poland, although Lithuania did not accept this decision and refused to arrange normal diplomatic relations with Poland. This was the situation when Zygmund was appointed to the chair of mathematics there. In 1939 Zygmund was drafted into the Polish army as a reserve officer at the start of the Second World War. Poland was quickly defeated and all resistance by the Poles had ceased by 6 October 1939. Zygmund returned to Vilnius but then, on 10 October 1939, Vilnius was returned to Lithuania.

In 1940 Zygmund escaped with his wife and son from German controlled territory to the United States. After a short time at the Massachusetts Institute of Technology he was appointed to Mount Holyoke College in 1940. He later spoke of the peacefulness and security that Mount Holyoke had brought to his family after the distress of their war-time experiences. He remained on the staff there until 1945 although he spent the year 1942-43 at the University of Michigan. In 1945 he was appointed to the University of Pennsylvania, then, at the invitation of Stone, to the University of Chicago in 1947 where he remained until he retired in 1980. In 1948 he visited South America where he met Calderón. Zygmund brought Calderón back to Chicago, where he obtained his doctorate under Zygmund's supervision not too long after his arrival. Not only did Zygmund have the role of Calderón's teacher and advisor, but was also his mentor and became his main collaborator. We discuss briefly below the most important topic of their collaboration.

Zygmund worked in analysis, in particular in harmonic analysis. The main topics he studied are listed in [2]:-
Antoni Zygmund was one of the greatest and most influential analysts of this century. Among other topics, he worked on summability of numerical series, summability of general orthogonal series, trigonometric integrals, sets of uniqueness, summability of Fourier series, differentiability of functions, smooth functions, approximation theory, absolutely convergent Fourier series, multipliers and translation invariant operators, conjugate series and Taylor series, lacunary trigonometric series, series of independent random variables, random trigonometric series, the Littlewood-Paley, Luzin and Marcinkiewicz functions, boundary values of analytic and harmonic functions, singular integrals, partial differential equations and interpolation operators.
Zygmund's early work, as one might expect, continued to develop ideas from his doctoral studies with Aleksander Rajchman. He studied topics such as Riemann summability, differentiability properties of trigonometric series and sets of uniqueness. For example in 1926 he published six papers in Mathematische Zeitschrift in French: Contribution à l'unicité du développement trigonométrique ; Sur la théorie riemannienne des séries trigonométriques ; Sur la possibilité d'appliquer la méthode de Riemann aux séries trigonométriques sommables par le procédé de Poisson ; Sur les séries trigonométriques sommables par le procédé de Poisson ; Sur un théorème de la théorie de la sommabilité and Une remarque sur un théorème de M Kaczmarz .

After meeting Paley at Cambridge in 1930-31, Zygmund wrote five joint papers with him, and both of them together with Norbert Wiener wrote Notes on random functions (1933). But for the tragic death of Paley in 1933 it is certain that this collaboration would have led to numerous further papers of major importance. The joint Zygmund-Paley work played an important role in Zygmund's book Trigonometric Series (1935). It is a classic that, together with later editions, is still the definitive work on the subject. A reviewer of the first edition of the book wrote:-
Each volume of the series [Monografie Mat.] published so far represents an important event in the development of mathematical research, and the present volume in this respect is second to none of its predecessors. If one looks through the long list of books on Fourier series one cannot help feeling that even the bulkiest of them are far from giving an adequate picture of the present status of the field. The non-existence of a monograph giving such a picture was very badly felt not only by beginners but also by specialists, and the failure of so many attempts to write a real book on Fourier series created an impression that the task was almost hopeless. The author of the present monograph completely succeeded in dispelling this "inferiority complex" and produced a book which not only introduces the reader into the immense field the theory of Fourier series but at the same time almost imperceptibly brings him to the latest achievements, many of them being due author himself. The style of the book is rigorous and vigorous and the exposition elegant and clear to smallest details.
A second edition appeared in 1959 published by Cambridge University Press. A reviewer wrote:-
In his course at the University of Cambridge, Professor Littlewood used to call the first edition of Zygmund's book "the Bible". This second edition, coming almost twenty-five years after the first one, will undoubtedly deserve this name even more, not only because it takes into account the work done in the field during this period, but also because the author, profiting from new experience and constant reflection on his past work, has introduced many topics which had been aside in the first edition
Remarkably a third edition appeared in 2002. This edition contains a foreword by Robert A Fefferman in which he writes:-
Surely, Antoni Zygmund's "Trigonometric series" has been, and continues to be, one of the most influential books in the history of mathematical analysis. Therefore, the current printing, which ensures the future availability of this work to the mathematical public is an event of major importance. Its tremendous longevity is a testimony to its depth and clarity. Generations of mathematicians from Hardy and Littlewood to recent classes of graduate students specializing in analysis have viewed "Trigonometric series" with enormous admiration and have profited greatly from reading it. In light of the importance of Antoni Zygmund as a mathematician and of the impact of "Trigonometric series", it is only fitting that a brief discussion of his life and mathematics accompany the present volume, and this is what I have attempted to give here. I can only hope that it provides at least a small glimpse into the story of this masterpiece and of the man who produced it.
Another major work, this time written with his long standing friend Saks, was Analytic functions published in 1938. An English edition appeared in 1952. Boas wrote:-
The authors have drawn on both the "geometric" and "arithmetic" approaches to the subject, and cover considerably more material than is usual in an introductory text. One of the novel features is the early introduction of Runge's theorem and its application to facilitate the proof of Cauchy's theorem and other results. Conformal mapping is discussed relatively late. The general part of the book is followed by three chapters, one on entire and meromorphic functions, one on elliptic functions, and one on $G(s), Z(s)$ and Dirichlet series.
In 1950, not long after beginning his long association with the University of Chicago, Zygmund published Trigonometric Interpolation. He described these as follows:-
The purpose of these notes is to present those aspects of trigonometric interpolation which resemble the theory of Fourier series.
Zygmund created one of the strongest analysis schools of the 20th century, making Chicago into a major analysis research centre (see [6] for details). In 1986 he received the National Medal for Science for building this research school. He supervised Nathan Fine at the University of Pennsylvania an over 40 doctoral students in his years at Chicago, including Alberto Calderón and Paul Cohen who have biographies in this archive.

We mentioned earlier in this article Zygmund's collaboration with Calderón. The main topic of their collaboration was the theory of singular integrals. Their joint work began with Calderón's doctoral thesis which answered some questions related to boundary behaviour of harmonic functions which had been posed by Zygmund earlier. Their famous joint papers over the next few years on singular integrals and partial differential equations, the most significant of which appeared in 1952, have had a major impact on modern analysis.

Together with Richard L Wheeden, Zygmund wrote Measure and integral (1977). A reviewer writes:-
This textbook has been conceived with the explicit intention of providing an easy and quick access to the most useful techniques of measure and integration in the modern analysis of real variables. This goal has been achieved with very remarkable success.
For outstanding contributions to Fourier analysis and its applications to partial differential equations and other branches of analysis, and for his creation and leadership of the strongest school of analytical research in the contemporary mathematical world.
Zygmund suffered many sad losses during his life. His doctoral student at Vilnius, Ela-Chaim Cunzer, was a victim of the Holocaust. His student and collaborator Marcinkiewicz died in 1940. Rajchman, Zygmund's doctoral supervisor, was executed by the Nazis. Saks, his friend from childhood and later his collaborator, was killed by the Gestapo. His young collaborator Paley tragically died in a mountaineering accident. Irena, his wife, died on 6 August 1966.

### References (show)

1. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Antoni-Zygmund
2. Antoni Szczepan Zygmund: December 26, 1900-May 30, 1992, Studia Math. 103 (2) (1992), 119-121.
3. R Askey, In memoriam: Antoni Zygmund (December 26, 1900-May 30, 1992), J. Approx. Theory 71 (1) (1992), 1-2.
4. A P Calderón, Antoni Zygmund, in Conference on harmonic analysis in honor of Antoni Zygmund, Chicago, Ill., 1981 1 (Wadsworth, Belmont, CA, 1983), xiii-xv.
5. A P Calderón and E M Stein, Antoni Zygmund 1900-1992, Notices Amer. Math. Soc. 39 (1992), 848-849.
6. R R Coifman and R S Strichartz, The school of Antoni Zygmund, in A century of mathematics in America III (Amer. Math. Soc., Providence, R.I., 1989), 343-368.
7. C Fefferman, J-P Kahane and E M Stein, The scientific achievements of Antoni Zygmund (Polish), Wiadomosci matematyczne (2) 19 (1976), 91-126.
8. G G Lorentz, Antoni Zygmund and his work, J. Approx. Theory 75 (1) (1993), 1-7.
9. E M Stein, Calderón and Zygmund's theory of singular integrals, in Harmonic analysis and partial differential equations, Chicago, IL, 1996 (Univ. Chicago Press, Chicago, IL, 1999), 1-26.
10. E M Stein, Singular integrals: the roles of Calderón and Zygmund, Notices Amer. Math. Soc. 45 (9) (1998), 1130-1140.
11. E M Stein, The development of square functions in the work of A Zygmund, in Conference on harmonic analysis in honor of Antoni Zygmund, Chicago, Ill., 1981 1 (Wadsworth, Belmont, CA, 1983), 2-30.
12. D Waterman, The contributions of Antoni Zygmund to real analysis, Real Anal. Exchange 19 (1) (1993/94), 11-12.
13. G Weiss, Antoni Zygmund 1900-1992, J. Geom. Anal. 3 (6) (1993), 529-531.
14. G Weiss, Antoni Zygmund 1990-1992, in Fourier analysis and partial differential equations, Miraflores de la Sierra, 1992 (CRC, Boca Raton, FL, 1995), 3-5.
15. A Zygmund, Notes on the history of Fourier series, in Studies in harmonic analysis, Proc. Conf., DePaul Univ., Chicago, Ill., 1974 (Math. Assoc. Amer., Washington, D. C., 1976), 1-19.
16. A Zygmund, The role of Fourier series in the development of analysis, Historia Math. 2 (4) (1975), 591-594.