Zygmund was enormously influential in the development of 20th-century mathematics, perhaps even more so through his students than directly. He had over 80 Ph D students, including Calderon himself, with whom he initiated and developed the study of Singular Integrals in a series of seminal works in the Fifties. The lynchpin of this theory is the so-called Calderon-Zygmund decomposition, still the deepest idea in modern Harmonic Analysis.
Amongst his other students who have gone on to be world leaders in their own right are Paul Cohen and Eli Stein. His second-generation mathematical descendants number many hundreds of active researchers across the globe. His book Trigonometric Series (1935 and 1959) is the classical and definitive treatment of the subject and is still required reading for every prospective student of hard analysis.
Zygmund was also a kind and generous man. Well into his eighties he would attend the analysis seminars at Chicago and offer deep and penetrating observations. But perhaps the maxim of his which will be the most remembered is 'when judging a mathematician's work, one should 'integrate only his f+ ' ' - Zygmund's way of saying that only positive contributions and not negative ones should be considered. With this in mind, one can only say that Zygmund's f+ was very very large indeed.