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Book review

Author(s) Kolmogorov, A.N.
Title Selected Works of A.N. Kolmogorov, Volume I: Mathematics and Mechanics
Publisher Kluwer Academic Publishers
Year of publication 1991
Reviewed by Borislav Bojanov

[This volume edited by V.M. Tikhomirov]

There is no other man in contemporary mathematics comparable with Kolmogorov by the power of his influence, the broadness of his interests, the wealth of his ideas, the number and fame of his students. Andrei Nikolaevich Kolmogorov was the last encyclopedist in the mathematical science. His extraordinary talent and activity has been creating the specific atmosphere at the Moscow State University during decades. So, I barely need to say more about the present book than what is given in the title: This is the first volume of the selected works of Kolmogorov on mathematics and mechanics.

The works are published in three volumes. The second volume is devoted to probability theory and mathematical statistics, and the third one to information theory and the theory of automata. The papers are selected and grouped by Kolmogorov himself.

The present volume contains 60 papers treating trigonometric and orthogonal series, theory of measure and integral, theory of functions, approximation theory, geometry, topology, turbulence and classical mechanics. All they are translated in English. The reader can find here the famous example of a 2pi-periodic function whose Fourier series is divergent everywhere (given by Kolmogorov at the age of nineteen), the story of the solution of Hilbert's thirteenth problem, his invited lecture in the International Congress of Mathematics in Amsterdam, 1954, the original proof of Kolmogorov inequality for the derivatives and many other results, which lie now in the foundations of new branches of mathematics. Most of the papers have been published originally in Russian and thus not easily available for the English speaking audience.

An essential part (more than 100 pages) of the book are the commentaries to the papers. Most of them are written by Kolmogorov and others by experts as Arnold, Uspenski, Arkhangelski, Yaglom, Tikhomirov, Ulyanov.

The volume starts with a brief biography of Kolmogorov written by his student V.M. Tikhomirov and ends with a complete list of works of the great master.

This is one of those books which should be found at every mathematical library. I recommend it to all mathematicians, students or specialist.