ANTONI ZYGMUND TRIGONOMETRIC SERIES PDF

He was born into a partitioned Poland, the third of the three partitions took place in and split Poland between Russia, Prussia and Austria. He attended elementary school in Warsaw, completing this stage of his education in After two years of secondary education World War I broke out in and Antoni, together with the rest of the Zygmund family, were evacuated to Poltava in the Ukraine. Antoni continued his education in Poltava then, in , he returned to Poland which had become an independent country for the first time in well over years.

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Warsaw, Poland, 26 December ; d. Chicago, Illinois, 30 May , mathematics, harmonic analysis, trigonometric series, singular integrals. He was a major figure in taking the subject from one to several variables and in the creation of a theory of singular integrals. Early Life and Career. On his return to Warsaw he found there was no opportunity to study astronomy, his first interest, so he switched to mathematics at the University of Warsaw. There he became attached to Aleksander Rajchman and Stanislaw Saks.

Hardy at Oxford University and John E. Littlewood at Cambridge University ; he also met and worked with Raymond E. Paley that year. Zygmund found the visit to England was a tremendous stimulus. He went on to write five papers with Paley and a joint paper with Paley and Norbert Wiener on lacunary and random series. There he met Jozef Marcinkiewicz, who was then a student but soon became a collaborator with Zygmund.

Even in the early s, Zygmund showed his distaste for manifestations of anti-Semitism, which eventually cost him his job in a politically motivated purge of the university in Hardy, Littlewood, Henri Lebesgue in France, and other eminent mathematicians wrote in protest, and Zygmund was reinstated.

It was during his time at Wilno that Zygmund wrote the first edition of his book Trigonometric Series This volume was so complete in its treatment that it was revised and reprinted by Cambridge University Press in and was reprinted six times, to become the standard work in its subject. The s were productive for Zygmund in many ways, but they ended in tragedy. Zygmund and Marcinkiewicz joined the Polish army, but the partition of Poland between the Nazis and the Soviets saw Wilno fall in the Russian zone.

Many of the Polish Officer Corps were rounded up by the Russians and massacred at Katyn in Poland, and most likely Marcinkiewicz was among them. Saks and Rajchman were murdered by the Nazis, and so in a brief and surely terrifying period of time, Zygmund lost most of his collaborators and close friends. In he transferred to the University of Pennsylvania in Philadelphia, and two years later he moved to the University of Chicago , where he stayed for the rest of his career.

All told, Zygmund had thirty-five students, many of whom went on to have distinguished careers. They included the National Medal of Science in , the highest honor awarded by the U. National Academy of Sciences and other national science academies, including those of Poland, Argentina, and Spain. Trigonometric Series. These had been introduced by Bernhard Riemann in as a significant generalization of the usual Fourier series.

As Zygmund learned from Rajchman, the key questions here concern the uniqueness of the series and its local properties such as continuity at a point. Uniqueness requires the nontrivial result that a trigonometric series that converges to zero everywhere has all of its coefficients zero and is therefore the trivial series. This leads to the study of sets E such that any trigonometric series that converges to zero outside E is necessarily the trivial series.

Such sets have measure zero, so their study requires new analytic tools, and Zygmund made a profound investigation of these sets. The book Trigonometric Series owes much to the influences of Saks and Marcinkiewicz. Integrable functions of one variable have an averaging property that is easy to generalize to functions of n variables.

However, Zygmund showed in , by using a construction of Otton Nikodym, that the generalization is false in dimensions higher than 1, and Saks then showed that even modest generalizations will fail. Zygmund was able to show, however, that the generalization can be made to work for functions of several variables that are in the class L p for some p. In later work with Marcinkiewicz, the class of functions for which the generalization holds was widened considerably. Single variable harmonic function theory is almost interchangeable with single variable complex function theory; indeed, that was the key insight of Riemann.

They used ideas of Marcinkiewicz as Zygmund was later to acknowledge and a number of powerful original ideas to establish the existence of the relevant singular integrals. Both the techniques and the results of this paper exerted a considerable influence on the future direction of work in this field. Singular Integrals. The integral operators in which they were interested have their roots in the classical theory of partial differential equations.

This allowed their theory to merge with the ideas of Russian mathematician Israel M. A profusion of work by many authors in many countries saw the theory of singular integrals become a major part of a much broader theory of what are called pseudo-differential operators.

Singular integrals, however, remain central to the study of real functions of several variables, and the work of Zygmund and his collaborators in the Chicago school of analysis decisively deepened that whole branch of mathematics. Trigonometric Series was first published in and in a third edition in The book is remarkable for both its thoroughness and its many highlights, among which is the Marcinkiewicz interpolation theorem, which was to play an important part in the creation of the theory of singular integrals.

It applies to operators of weak type, and singular integrals on the space L 1 are of weak type. Zygmund also drew on his time with Hardy, Littlewood, and Paley in writing the book, and the Hardy-Littlewood maximal theorem is central to it. It led Zygmund to deepen the approach to Fourier series of a single variable by using complex variable methods, and it has implications for the study of the Hardy spaces H p, all of which are described in the book.

Zygmund himself had a particular liking for the material on the Littelwood-Paley functional, which applied to a function produces a new function with an L p norm comparable in size with the L p norm of the original function. This makes it very useful, and Zygmund used it to study Hardy spaces.

Selected Papers of Antoni Zygmund. Dordrecht, Netherlands; Boston: Kluwer Academic, Trigonometric Series, 3rd ed. Cambridge, U.

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