[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 in 1986.

In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts.

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".

[6] Zygmund's students included Alberto Calderón, Paul Cohen, Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar.