#### by Richard Askey

Antoni Zygmund died on May 30, 1992, at the age of 91. His mathematical legacy lives on, in his papers, his books, and his students and their students.

His research includes such major work as the theory of singular
integrals in __\( n \)__-dimensions which goes under the name of
*Calderón and
Zygmund singular integrals*. He was one of the first to realize the
importance of complex methods in harmonic analysis and, independently
of
Thorin, discovered a complex variable proof of the
M. Riesz
interpolation theorem. He also wrote the first detailed treatment of
Marcinkiewicz’s
interpolation theorem. These are the two main tools
which lead to the theory of interpolation spaces, both the complex
theory and the real theory. In approximation theory, he showed that
__\( L \log L \)__ is the right substitute for __\( L \)__ in many questions, and he
introduced the space which he called __\( \Lambda \)__ as the right space to
substitute for __\( \mathit{Lip}(1) \)__, and then proved one of the early
saturation theorems. Many of his other contributions have been used by
others in approximation theory.

It is difficult for younger mathematicians to realize the importance of the
first edition of Zygmund’s *Trigonometric Series*. It was thought at the time
when he wrote this book that it would be impossible to bring order into
the complicated results which seemed not to have an overriding structure.
I heard this from more than one person when I started to read this book,
twenty years after it was first published. After the second edition was
published, I never heard *Fourier Analysis* described as a subject without
structure, since *it was there* in these two marvelous volumes.

As important as Zygmund’s research, his books, and his two joint
texts on complex and real analysis were, his work with students is
probably his most important legacy. The Calderón–Zygmund seminar, as
it was known when I went to it for two years, was an exciting place to
learn how mathematics was done, how problems are uncovered and
developed, and what mathematical taste is. There was a one hour
lecture, with Zygmund often making very perceptive comments, and then
an informal second hour which was full of open questions, some of
which would be solved on the spot by others in the room. It played a
central role in my development as a mathematician, and for many others
as well. A list of Zygmund’s students in the United States and their
students is given in Part III of *A Century of Mathematics in America*.1
Zygmund also had Ph.D. students in Poland, including
J. Marcinkiewicz. Zygmund’s account of how Marcinkiewicz developed appears in the
*Collected Papers* of Marcinkiewicz which Zygmund edited. The *Selected
Papers of Antoni Zygmund* 2
should be in every major library, and the
account of Zygmund’s work contained there should be read by people who
want to know about some of the impact Zygmund had on the development
of mathematics in this century.

Zygmund was a gentle man, always looking for the best in others. To
quote from the paper on his students,3
Zygmund told one of his
students, “Concentrate only on the achievements, and ignore the
mistakes. When judging a mathematician you should only integrate
__\( f_{+} \)__ (the positive part of his function) and ignore the negative
part.” This is wise advice from a very wise man.