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**Selected papers of Antoni Zygmund. Volume 1-3. Ed. by A. Hulanicki, P. Wojtaszczyk and W. Zelazko.**
*(English)*
Zbl 0689.01020

This careful edition contains photographic reprints of 79 papers out of 215, summed up in the bibliography of Zygmund’s works. The bibliography, which is repeated in each volume, contains useful reference to reviews of Zygmund’s articles in Zentralblatt and Mathematical Reviews. Antoni Zygmund (born 26 December 1900), one of the brilliant Polish mathematicians of the 20s and 30s of our century, did influential research work especially in the theory of trigonometric series, his treatise on this subject (1935, second ed. 1959) being one of the monuments of the century in mathematics. W. Zelasko’s short biographical note (vol. 1, XII-XV, with a photograph of Zygmund’s at p. XII) traces Zygmund’s career back to his study with his teacher A. Rajchman in Warsaw, to his early collaboration with R. Salem in Paris (1928) and R. Paley in Cambridge (1930). After the Polish army’s defeat, Zygmund left under compulsion his chair in Wilno in 1940, where he had fruitful collaboration with his student J. Marcinkiewsicz before. [Joint papers with Marcinkiewicz are excluded from this edition, since they are included in J. Marcinkiewicz’s “Collected papers” (Warsaw 1964).] Zygmund finally settled in Chicago, where he had 35 Ph. D. students out of 39 totally, which are listed up in the first volume (XVI/XVII). This volume contains also “A review of A. Zygmund’s scientific work” (XIX- XLVIII), written by Ch. Fefferman, J.-P. Kahane and E. M. Stein and originally published in Polish in 1976 [Wiadom. Mat. 19, 91-126 (1976; Zbl 0328.01019)]. This article gives emphasis to the breadth of Zygmund’s interests, ranging from functions of one and several real variables, complex analysis and functional analysis in probability theory. Among other results Zygmund’s notion of smooth functions (1945) is mentioned, a crucial concept in real analysis. Zygmund’s use (together with A. P. Calderón 1952) of singular integrals proved to be “one of the most powerful tools in partial differential equations” (XLIII). The authors stress Zygmund’s indirect influence on mathematics through his students and Zygmund’s appreciation of the results of other mathematicians. This last fact becomes evident from Zygmund’s historical articles, which are not included in this valuable edition.

Reviewer: R.Siegmund-Schultze

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |