#### by Daniel Waterman

At the beginning of Zygmund’s career he wrote a few papers on real analysis which were not directly related to trigonometric series. It is interesting to note, however, that these papers and the few such that he wrote later in life were always joint, the problems presumably arising from a confluence of interests of the parties.

Typical of these are the paper of 1923 with
Sierpiński showing that
there exist functions discontinuous on every set having the power of
the continuum and another with
Saks in 1924 on the sheaf of tangent
lines from a point to a planar curve.1
Another such result appears in Saks’s *Theory of the Integral*,
noted (in the original French version) as Zygmund’s Lemma, and used in
the proof of the Fundamental Theorem of the Calculus for the Perron
integral.

Most of Zygmund’s contributions are subsumed under the rubric of
“differentiation theory”, which is a natural consequence of his
primary interests in trigonometric series. The sequence of partial
sums __\( {s_n(x)} \)__ of the trigonometric Fourier series of a function __\( f \)__
is transformed by a summability method into a sequence __\( \{\sigma_n(x)\} \)__
and the difference __\( \sigma_n (x)-f(x) \)__ is represented by the
convolution of a symmetric difference and a summability kernel. The
simplest example is the case where the method is convergence itself
and __\( \sigma_n(x)-f(x) \)__ is essentially the convolution of
__\[ f(x+t)+f(x-t)-2f(x) \qquad\text{and}\qquad (\sin nt)/t .\]__
Similarly, his work in 1934
and his joint paper in 1935 (with
Jessen and
Marcinkiewicz) on the
differentiability of multiple integrals arose from problems in the
summability of double Fourier series.

In the study of trigonometric series, the conjugate series is of great
importance; in the case of the Fourier series of a function, the
associated conjugate function was the center of much of Zygmund’s
work. In 1929 he extended the result of
M. Riesz on the integrability
of the conjugate function from the class __\( L^p \)__, __\( p > 1 \)__, to the class
__\( L\log L \)__, which proved to be an essential ingredient of much of his
later work. In the same paper he also introduced the spaces
__\( L_{\varphi} \)__, which are now called Orlicz spaces although the paper of
Orlicz appeared in 1931. He explored the relationships between many
derivatives; the Peano, Schwarz, Borel, Riemannian and approximate
derivatives were all considered. In a particularly deep paper of 1936
with Marcinkiewicz, although most of the paper dealt with
differentiation, he defined the T integral of Perron type. With the T
integral he solved a problem of
Denjoy by showing that if a
trigonometric series converges everywhere to a function __\( f \)__, then the
series is the T-Fourier series of __\( f \)__.

The notion of smoothness which appears in 1945 is a generalization of
symmetric continuity and __\( \Lambda_{\ast} \)__ is a generalization of the
class of integrals of __\( \mathit{Lip}(1) \)__ functions. In this paper he also studies
fractional integrals and derivatives. Many of these results were
generalized to __\( n \)__-dimensions with
Calderón in 1954, and the notion
of smoothness is applied to explain a result of
Salem in a paper with
Mary Weiss in 1959. He returned to the basic questions in the 1945
paper in a series of deep papers with
E. M. Stein in 1961, 1964 and 1965.

In the 1952 paper with Calderón on the existence of singular integrals, a covering lemma is of critical importance. This lemma provides the basis for the decomposition of a function into its “good” and “bad” parts. These techniques are featured in several papers published with Calderón in the 50’s and early 60’s. The most notable may be the 1961 paper on local properties of solutions of elliptic PDE’s.