Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. He worked mostly in the area of

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, including

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

, and he is considered one of the greatest analysts of the 20th century. 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 National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavior ...

in 1986.

Biography

Born in

Warsaw Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at ...

, Zygmund obtained his Ph.D. from the

University of Warsaw The University of Warsaw (, ) is a public university, public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well ...

(1923) and was a professor at Stefan Batory University at

Wilno Vilnius ( , ) is the capital of and List of cities in Lithuania#Cities, largest city in Lithuania and the List of cities in the Baltic states by population, most-populous city in the Baltic states. The city's estimated January 2025 population w ...

from 1930 to 1939, when

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

broke out and Poland was occupied. In 1940 he managed to emigrate to the

United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...

, where he became a professor at

Mount Holyoke College Mount Holyoke College is a Private college, private Women's colleges in the United States, women's Liberal arts colleges in the United States, liberal arts college in South Hadley, Massachusetts, United States. It is the oldest member of the h ...

South Hadley, Massachusetts South Hadley (, ) is a New England town, town in Hampshire County, Massachusetts, United States. The population was 18,150 at the 2020 United States census, 2020 census. It is part of the Springfield metropolitan area, Massachusetts. South Hadle ...

. In 1945–1947 he was a professor at the

University of Pennsylvania The University of Pennsylvania (Penn or UPenn) is a Private university, private Ivy League research university in Philadelphia, Pennsylvania, United States. One of nine colonial colleges, it was chartered in 1755 through the efforts of f ...

, and from 1947, until his retirement, at the

University of Chicago The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...

. He was a member of several scientific societies. From 1930 until 1952 he was a member of the

Warsaw Scientific Society Warsaw Scientific Society (Polish: ''Towarzystwo Naukowe Warszawskie''; TNW) is a Polish scientific society based in Warsaw Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city o ...

(''TNW''), from 1946 of the

Polish Academy of Learning The Polish Academy of Arts and Sciences or Polish Academy of Learning (, PAU), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of sciences (the other being the Po ...

(''PAU''), from 1959 of the

Polish Academy of Sciences The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars a ...

(''PAN''), and from 1961 of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

in the United States. In 1986 he received the

. In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume ''

Trigonometric Series In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynom ...

''. 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".The 2nd edition of Zygmund's ''Trigonometric Series'' (

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, 1959) consists of 2 separate volumes. The 3rd edition (Cambridge University Press, 2002, ) 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. Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of

trigonometric series In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynom ...

had remained the largest component of Zygmund's mathematical investigations. His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the results he obtained with Calderón on singular integral operators. George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work". Zygmund's students included Alberto Calderón,

Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a F ...

, Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar. He died in

Chicago Chicago is the List of municipalities in Illinois, most populous city in the U.S. state of Illinois and in the Midwestern United States. With a population of 2,746,388, as of the 2020 United States census, 2020 census, it is the List of Unite ...

Family

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

Mathematical objects named after Zygmund

Calderón–Zygmund lemma In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where den ...

* Marcinkiewicz–Zygmund inequality * Paley–Zygmund inequality * Calderón–Zygmund kernel

Books

*Trigonometric Series (Cambridge University Press 1959, 2002) *Intégrales singulières (

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 1971) *Trigonometric Interpolation (University of Chicago, 1950) *Measure and Integral: An Introduction to Real Analysis, With Richard L. Wheeden (

Marcel Dekker Marcel Dekker was a journal and encyclopedia publishing company with editorial boards found in New York City. Dekker encyclopedias are now published by CRC Press, part of the Taylor and Francis publishing group. History Initially a textbook publ ...

, 1977) *Analytic Functions, with Stanislaw Saks (

Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, ...

Science Ltd, 1971)

References

External links

*
Mount Holyoke biography
* {{DEFAULTSORT:Zygmund, Antoni 1900 births 1992 deaths People from Warsaw Governorate 20th-century Polish mathematicians Mathematical analysts National Medal of Science laureates Members of the Polish Academy of Learning University of Warsaw alumni Mount Holyoke College faculty University of Pennsylvania faculty University of Chicago faculty Academic staff of Vilnius University Polish emigrants to the United States Functional analysts Members of the Polish Academy of Sciences Members of the United States National Academy of Sciences