Biography:Antoni Zygmund

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Antoni Zygmund
Zygmund, ca. 1935
Born(1900-12-26)December 26, 1900
Warsaw, Congress Poland, Russian Empire
DiedMay 30, 1992(1992-05-30) (aged 91)
Chicago, Illinois, United States
CitizenshipPolish, American
EducationUniversity of Warsaw (Ph.D., 1923)
Known forSingular integral operators
Calderón–Zygmund lemma
Marcinkiewicz–Zygmund inequality
Paley–Zygmund inequality
Calderón–Zygmund kernel
Chicago school of mathematical analysis
AwardsLeroy P. Steele Prize (1979)
National Medal of Science (1986)
Scientific career
FieldsMathematics
InstitutionsUniversity of Chicago
Stefan Batory University
Doctoral advisorAleksander Rajchman
Stefan Mazurkiewicz
Doctoral studentsAlberto Calderón
Paul Cohen
Mischa Cotlar
Nathan Fine
Józef Marcinkiewicz
Benjamin Muckenhoupt
Stylianos Pichorides
Victor L. Shapiro
Elias M. Stein
Guido Weiss

Antoni Szczepan Zygmund (pl; 26 December 1900[1] – 30 May 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century.[2][3][4][5][6] Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science, the highest American science award, in 1986.[2][3][4][5]

Life and career

Zygmund was born on 26 December 1900 in Warsaw, then part of the Russian Empire, in a Polish family of Wincenty Zygmund and Antonina (née Perowska).[7] He was the oldest of four siblings, with three younger sisters. In 1914, his secondary education in Warsaw was interrupted by the outbreak of World War I forcing his family to relocate to Poltava (presnet-day Ukraine). In 1918, he returned to Poland, which had just become an independent state, and spent a year studying at Kazimierz Kulwiee’s Gymnasium in Warsaw.[7]

The following year, he enrolled at the University of Warsaw[8] and studied under prominent mathematicians representing the Warsaw School of Mathematics including Zygmunt Janiszewski, Stefan Mazurkiewicz and Wacław Sierpiński.[9] In 1920, Zygmund’s education was interrupted again when he was drafted into the army. He did not take part in active combat in the Polish–Soviet War and was therefore able to resume his academic work. During his time at the University of Warsaw, Zygmund was particularly influenced by Aleksander Rajchman, attending his lectures and a seminar on the theory of trigonometric series. In 1923, Zygmund obtained his doctorate from the University of Warsaw based on his thesis concerning the Riemannian theory of trigonometric series written under the formal supervision of Mazurkiewicz.[7][9]

In 1922–26, Zygmund worked as an assistant at the Warsaw University of Technology. In 1926, he earned his habilitation and assumed the position of a docent at the University of Warsaw. In 1929, he was granted a Rockefeller Foundation scholarship[10] and spent a year in Oxford and Cambridge, which exerted a profound impact upon his scientific development. He collaborated with such mathematicians as Raymond Paley,[11] G.H. Hardy and J. E. Littlewood, publishing five joint-papers with Paley between 1930 and 1931.[8][7]

In 1935, Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field".[12] Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst.[6] In 1988, a new edition of the work was published by Cambridge University Press.[13] The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations.[6]

Zygmund also worked with Stanisław Saks, with whom he co-authored the book Analytic Functions published in the Monografie Matematyczne (Mathematical Monographs) series in 1938.[8]

Zygmund was a professor at Stefan Batory University in Wilno from 1930 to 1939. During this period, he began a highly productive scientific collaboration with his most gifted student, Józef Marcinkiewicz, with whom he also developed a close personal friendship. Together they obtained significant results in real and Fourier analysis, probability theory, and moment inequalities, contributions that later became foundational in several areas of modern mathematical analysis.[14][15] Zygmund also further developed many of Marcinkiewicz's ideas after his student's tragic death and played an important role in helping preserve his scientific legacy abroad.[8][14]

Zygmund spent most of his academic career at the University of Chicago

When World War II broke out, Poland was occupied and, in 1940, Zygmund managed to emigrate to the United States. Thanks to the support from Jerzy Neyman, Jacob Tamarkin and Norbert Wiener, he received an invitation to Mount Holyoke College in South Hadley, Massachusetts where he became a professor.[8] In 1945–1947, he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. During his time in Chicago, Zygmund together with his student Alberto Calderon (whom he had met in 1948 in Argentina),[9] founded the Chicago school of mathematical analysis, a school of thought in mathematics that highlights the use of Fourier analysis in the study of partial differential equations. It became one of the most influential analytical schools of the 20th century, contributing many of the field’s major advances.[16][17][18]

He was a member of several scientific societies. From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States.[7]

In 1954, Zygmund participated as an invited speaker at the International Congress of Mathematicians in Amsterdam.[10]

In 1979, Zygmund was awarded the Leroy P. Steele Prize from the American Mathematical Society for "his cumulative influence on the theory of Fourier series, real variables, and related areas of analysis."[19]

In 1986, Zygmund received the National Medal of Science, the highest science award in the United States,[20][21] from President Ronald Reagan for his outstanding contribution to the development of Fourier analysis, as well as its application to partial differential equations and other branches of mathematical analysis.[22] He was also recognized for his "creation and leadership of the strongest school of analytical research in the contemporary mathematical world."[7][16][17][18]

In 1981, he was granted emeritus status.[13] Zygmund continued attending analysis seminars at Chicago even in his eighties, contributing profound and incisive insights.[23] He died on 30 May 1992 in Chicago, at the age of 91.[13]

Zygmund published six books and more than 180 scientific papers during his career.[13] He was the advisor of 40 doctoral students, including Alberto Calderón, Paul Cohen (Fields Medal winner), Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar.[24]

Contributions

His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among his chief interests in that area was expressing vibrating bodies in precise mathematical terms, a key issue in spacecraft engineering, crystallography, and laser holography.[13]

Some of his most important contributions concerned the theory of singular integral operators, and were achieved in collaboration with Calderón.[23][12] In particular, the Calderón–Zygmund lemma is regarded as a fundamental result in this area of mathematics. It was developed to prove the weak-type continuity of singular integrals of integrable functions becoming a standard tool in analysis and probability theory. George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work".[6]

Zygmund, together with Raphaël Salem, demonstrated that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a Pisot number, that is an algebraic integer characterized by the fact that all its conjugates (if any) are smaller than 1. Their result was the first to show that the uniqueness property depends on arithmetic features rather than merely on notions of size.[25]

The Marcinkiewicz–Zygmund inequality is another significant result by Zygmund. It establishes relationships between the moments of independent random variables and constitutes a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order.

Other concepts named after Zygmund include the Paley–Zygmund inequality,[26] Calderón–Zygmund kernel, Calderón–Zygmund decomposition and Calderón–Zygmund operators.

Family

Antoni Zygmund, who had three sisters, married Irena Parnowska, a mathematician, in 1925.[4] Upon his death he was survived by four grandsons.[27]

Awards and recognition

Zygmund received a number of awards and distinctions, some of which include:[7]

  • 1953: Guggenheim Fellowship[10]
  • 1954: Invited Speaker at the International Congress of Mathematicians[10]
  • 1959: Member of the Polish Academy of Sciences
  • 1961: Member of the National Academy of Sciences
  • 1964: Member of the National Academy of Sciences of Argentina
  • 1967: Member of the Spanish Royal Academy of Sciences, the American Academy of Arts and Sciences, and the London Mathematical Society
  • 1972: Jurzykowski Foundation Prize, member of the Polish Mathematical Society
  • 1979: Leroy P. Steele Prize
  • 1986: National Medal of Science

He accepted honorary doctorates from Mount Holyoke College, Washington University[disambiguation needed], the Nicolaus Copernicus University in Toruń,[28] the University of Paris, and the University of Uppsala.[7]

Books

  • Trigonometric Series (Cambridge University Press 1959, 2002)
  • Intégrales singulières (Springer-Verlag, 1971)
  • Trigonometric Interpolation (University of Chicago, 1950)
  • Measure and Integral: An Introduction to Real Analysis, with Richard L. Wheeden (Marcel Dekker, 1977)
  • Analytic Functions, with Stanislaw Saks (Elsevier Science Ltd, 1971)

See also

References

  1. ↑ The Editors of Encyclopaedia Britannica, Antoni Zygmund at the Encyclopædia Britannica
  2. ↑ 2.0 2.1 Noble, Holcomb B. (1998-04-20). "Alberto Calderon, 77, Pioneer Of Mathematical Analysis" (in en-US). The New York Times. ISSN 0362-4331. https://www.nytimes.com/1998/04/20/us/alberto-calderon-77-pioneer-of-mathematical-analysis.html. 
  3. ↑ 3.0 3.1 Warnick, Mark S. (19 April 1998). "ALBERTO CALDERON, MATH GENIUS" (in en-US). https://www.chicagotribune.com/news/ct-xpm-1998-04-19-9804190439-story.html. 
  4. ↑ 4.0 4.1 4.2 "Antoni Zygmund (1900-1992)". http://www-history.mcs.st-and.ac.uk/Biographies/Zygmund.html. 
  5. ↑ 5.0 5.1 "PROFESSOR ALBERTO CALDERON, 77, DIES" (in en-US). Washington Post. ISSN 0190-8286. https://www.washingtonpost.com/archive/local/1998/04/20/professor-alberto-calderon-77-dies/069850e0-284c-4f03-9ce8-64a7e114274c/. 
  6. ↑ 6.0 6.1 6.2 6.3 Lorentz, G. G. (1993). "Antoni Zygmund and His Work". Journal of Approximation Theory 75: 1–7. doi:10.1006/jath.1993.1084. https://history-of-approximation-theory.com/fpapers/zygmund_work.pdf. 
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 J. J. O'Connor, E. F. Robertson. "Antoni Szczepan Zygmund". MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Biographies/Zygmund/. 
  8. ↑ 8.0 8.1 8.2 8.3 8.4 Wiesław Wójcik. "Zygmund Antoni Szczepan" (in pl). https://gigancinauki.pl/gn/biogramy/83777,Zygmund-Antoni-Szczepan.html. 
  9. ↑ 9.0 9.1 9.2 Marcin Bójko (19 April 2019). "Po co ci sinus? Poznaj Antoniego Zygmunda - matematyka, bez którego świat wyglądałby inaczej" (in pl). Gazeta Wyborcza. https://wyborcza.pl/AkcjeSpecjalne/7,160474,24678902,po-co-nam-sinus-i-cosinus.html. 
  10. ↑ 10.0 10.1 10.2 10.3 Editorial Committee of Studia Mathematica (1992). "Antoni Zygmund". Celebratio Mathematica. https://celebratio.org/Zygmund_A/article/403/. 
  11. ↑ Paley, R. E. A. C.; Zygmund, Antoni (1932). "A note on analytic functions in the unit circle". Proceedings of the Cambridge Philosophical Society 28 (3): 266–272. doi:10.1017/S0305004100010112. Bibcode: 1932PCPS...28..266P. 
  12. ↑ 12.0 12.1 The 2nd edition of Zygmund's Trigonometric Series (Cambridge University Press, 1959) consists of 2 separate volumes. The 3rd edition (Cambridge University Press, 2002, ISBN 0 521 89053 5) consists of the two volumes combined with a foreword by Robert A. Fefferman. The nine pages in Fefferman's foreword (biographic and other information concerning Zygmund) are not numbered.
  13. ↑ 13.0 13.1 13.2 13.3 13.4 Wolfgang Saxon (3 June 1992). "Antoni Zygmund, 91, Professor Of Mathematics in Chicago, Dies". The New York Times. https://www.nytimes.com/1992/06/03/us/antoni-zygmund-91-professor-of-mathematics-in-chicago-dies.html. 
  14. ↑ 14.0 14.1 "Wybraniec bogów -fabularyzowany film dokumentalny [English subtitles"]. IPNtvPL. 22 April 2022. https://www.youtube.com/watch?v=54VF6nzHKg8&t=3783s. 
  15. ↑ Roman Sznajder (19 March 2019). "Józef Marcinkiewicz–his life and work". norbertwiener.umd.edu. https://norbertwiener.umd.edu/NW/Marcinkiewicz_Bio_2.pdf. 
  16. ↑ 16.0 16.1 Noble, Holcomb B. (1998-04-20). "Alberto Calderon, 77, Pioneer Of Mathematical Analysis" (in en-US). The New York Times. ISSN 0362-4331. https://www.nytimes.com/1998/04/20/us/alberto-calderon-77-pioneer-of-mathematical-analysis.html. 
  17. ↑ 17.0 17.1 Warnick, Mark S. (19 April 1998). "ALBERTO CALDERON, MATH GENIUS" (in en-US). https://www.chicagotribune.com/news/ct-xpm-1998-04-19-9804190439-story.html. 
  18. ↑ 18.0 18.1 "PROFESSOR ALBERTO CALDERON, 77, DIES" (in en-US). Washington Post. ISSN 0190-8286. https://www.washingtonpost.com/archive/local/1998/04/20/professor-alberto-calderon-77-dies/069850e0-284c-4f03-9ce8-64a7e114274c/. 
  19. ↑ "Leroy P. Steele Prize (1970 - 1992)". American Mathematical Society. https://www.ams.org/prizes-awards/pabrowse.cgi?parent_id=28. 
  20. ↑ "About the National Medal of Science". https://www.nsf.gov/news/special_reports/medalofscience50/about.jsp. 
  21. ↑ "President Biden awards National Medal of Science to Brown engineer Subra Suresh" (in en). 2024-09-11. https://www.brown.edu/news/2023-10-24/suresh-medal-science. 
  22. ↑ Ludwika Tomala (10 March 2019). "Prof. Strzelecki: Antoni Zygmund to zdobywca matematycznych Himalajów" (in pl). https://naukawpolsce.pl/aktualnosci/news%2C33102%2Cprof-strzelecki-antoni-zygmund-zdobywca-matematycznych-himalajow.html. 
  23. ↑ 23.0 23.1 Carbery, Tony (17 July 1992). "Obituary: Professor Antoni Zygmund". The Independent. https://www.independent.co.uk/news/people/obituary-professor-antoni-zygmund-1533716.html. 
  24. ↑ "Antoni Zygmund". Mathematics Genealogy Project. https://genealogy.math.ndsu.nodak.edu/id.php?id=6388. 
  25. ↑ Bertin, M. J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; Schreiber, J. P. (1992). "The Salem–Zygmund Theorem". Pisot and Salem Numbers. Basel: Birkhäuser. doi:10.1007/978-3-0348-8632-1_15. 
  26. ↑ Ghosh, B. K. (2002). "Probability inequalities related to Markov's theorem". The American Statistician 56 (3): 186–190. doi:10.1198/000313002119. 
  27. ↑ "Obituary. Antoni Zygmund, 91, U. of C. Math Professor". Chicago Tribune. June 2, 1992. https://www.chicagotribune.com/1992/06/02/antoni-zygmund-91-u-of-c-math-professor/. 
  28. ↑ "Doktorzy honoris causa" (in pl). Nicolaus Copernicus University in Toruń. https://www.umk.pl/o-uczelni/historia-i-liczby/wyroznienia-uniwersyteckie/doktorzy-honoris-causa/. 

Further reading



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