# Anatoli Georgievich Vitushkin

Born
25 June 1931
Moscow, Russia
Died
9 May 2004
Moscow, Russia

### Biography

Anatoli Georgievich Vitushkin was the son of Egor Yegorovich Vitushkin and Pelageya Efimovna Vitushkin. In 1943 he was admitted to the Tula Suvorov Military School. He wrote [6]:-
Such military schools were established during the war, in 1943, in order to revive the old traditions of the former Russian Cadet Corps. A cadet, training to be an army officer in future, was supposed to get a thorough education and a proper up-bringing. In addition to the usual high school curriculum, the Suvorov School offered such subjects as the history of diplomacy, dancing, and higher mathematics.
He took part in the first victory parade on 24 June 1945 which celebrated the defeat of Germany in World War II. In 1946, as a result of an accident, he lost his sight. This fact seems hardly mentioned in a number of biographies, a fact which we should realise shows how successful Vitushkin was in not allowing this severe disability to affect his life and career. In [6] he described the teaching of mathematics at the Tula Suvorov Military School:-
Here is an example of how we studied higher mathematics. Lieutenant Georgii Ivanovich Bobylev taught mathematics to our platoon. I see now that he was an excellent teacher. His lessons were fascinating and joyful. Once we were not able to differentiate an intricate expression. Calling us slackers, he started explaining the rules of differentiation once again. While he was writing on the blackboard and we were listening to him quietly, our platoon joker, Gena Emel'yanov, purposely whispered so that everyone in the class was able to hear: "Engels said that even a monkey can be trained to differentiate." "But I also can integrate," replied the teacher, thus accepting the joke. By the end of the semester, announcing the grades, he recalled this monkey. While reading out Gena's grades he said: "Emel'yanov - four, four, four," then with a smile he counted the grades: "One, two, three, hence Emel'yanov gets 'three'." Of course, the genuine grade was "four."
In 1949 he graduated from Tula Suvorov Military School with a gold medal and, later that year, entered the Faculty of Mechanics and Mathematics of Moscow State University. At Moscow State University, he was a Stalin scholarship holder. He wrote [6]:-
Several of the professors in the department, including Kronrod, were looking at Hilbert's 13th problem at this time. In particular Andrei Nikolaevich Kolmogorov suggested that it was a good problem for students to work on. Vitushkin became interested in the problem and, while he was still an undergraduate, he had produced several contributions to this problem. In particular in 1954 he published five papers in Russian: Sufficient conditions for the boundedness of the linear variation of a function of three variables; Some estimates for variations of sets; On Hilbert's thirteenth problem; Determination of the variations of a set and a metric duality law; and Variations of functions of several variables and sufficient conditions for their boundedness.

He was already learning other important topics in his final undergraduate year [6]:-
In the fifth year at the university, having listened to Pontryagin's course on continuous groups, I was attracted to topology. For a year I attended the Seminar of Pavel Sergeevich Aleksandrov.
Vitushkin graduated from the university with honours in 1954 and became a post-graduate student of Andrei Nikolaevich Kolmogorov. In his work on Hilbert's 13th problem he had devised a concept of the variation of a function which generalised a notion already introduced by Kronrod. Remarkably he published a 220 page book on these ideas which appeared in print in the year after he had been awarded his first degree. This book is On multidimensional variations (Russian) (1955) which was reviewed by Laurence C Young who begins his review as follows:-
The central theme of this tract, a theme with which students of convex figures have long been familiar, is the desirability of attaching to a subset of Euclidean $n$-space, and also to a function of $n$ real variables, a string of $n + 1$ quantities $v_{k} (k = 0, 1, ... , n)$, whose values provide suitably independent items of useful information about the set, or about the function. The classical notions of $k$-dimensional measure of a set, and of $k$-dimensional total variation of a function, are inadequate in this respect ... In regard to the main theme of the tract, the particular definitions of the $v_{k}$ adopted by the author are natural and interesting, and they are reasonably motivated by their applications. ...
In 1957 Vitushkin submitted his thesis Variations of functions of several variables and sufficient conditions for their boundedness (Russian) for his Candidate's degree (equivalent to a Ph.D.). Beloshapka writes in [3]:-
Vitushkin's interest in similar themes [to his 1955 book] did not only have an intrinsically mathematical basis. He was enthusiastically and quite professionally concerned with problems of electrical engineering .... His household musical apparatus was for many years an object constantly in a state of being perfected and an object of pride. This gave rise to an interest in problems of sound recording. During his university years he took piano lessons and achieved moderate success. Electrical engineering, sound recording, and a passion for music all contributed to an interest in the problem of coding and transmission of signals. This interest was useful to him when, in 1957, after completing his graduate course work, Vitushkin began to work in the Institute of Applied Mathematics of the Academy of Sciences, where he was concerned not only with pure science but also with technological developments.
Vitushkin, in addition to working for his doctorate at Moscow State University, also joined the Institute of Applied Mathematics with a group of engineers, and they worked on the mathematical side of the space programme. Vitushkin was awarded a doctorate (about the level of the habilitation) by Moscow State University in 1958 for his dissertation On the difficulty of the tabulation problem (Russian). He published the 228-page book Estimation of the complexity of a tabulation problem (Russian) (1959) containing the main results of his doctoral thesis [4]:-
One of the central results of the doctoral dissertation and the monograph is a lower bound for the complexity of an algorithm of a general form that realizes an approximation to the elements of a compact function space ...
There was, however, a difficulty which Vitushkin did not like, namely the secret nature of the work at the Institute of Applied Mathematics meant that professional contacts were severely limited. He therefore transferred to the Steklov Mathematical Institute in 1965. He writes about an earlier contact with the Steklov Mathematical Institute in [6]:-
Andrei Nikolaevich [Kolmogorov] advised me to take a post-graduate course at the Steklov Mathematical Institute so that later on I would be able to get a research position there. On Kolmogorov's recommendation, Ivan Matveevich Vinogradov invited me for an interview. The interview was going smoothly until Vinogradov learned that I was Kronrod's student. The conversation then took an undesirable turn. Serge Mikhalovich Nikolski, who was also present there, tried to support me. However, Ivan Matveevich cut short my visit, giving his toothache as an excuse. When I left the room he said to Nikolski: "Never mind. He has nowhere else to go if not to us." [And] it did turn out his way. I became a post-graduate at the university and got a position at the Institute only eleven years later. This example shows that Vinogradov's position towards Jews left much to be desired. In other aspects he was an adequate head of the Institute and much respected, especially for his straightforwardness and consistency.
Also in 1965 Vitushkin began lecturing at Moscow State University. He was first appointed as a lecturer in the Department of Theory of Functions and Functional Analysis of the Faculty of Mechanics and Mathematics but was rapidly promoted reaching full professor in 1971. He was involved with the Complex Analysis Seminar for nearly 40 years, and for the last 10 to 15 of these years he directed the Seminar.

In 1974 Vitushkin was a plenary speaker at the International Congress of Mathematicians held in Vancouver, Canada in August of that year. He delivered the lecture Coding of Signals with Finite Spectrum and Sound Recording Problems. We give the first few sentences of his lecture:-
We will discuss one well-known problem of information theory, the problem which at present arises in various branches of radio engineering. We mean the problem of coding signals with finite spectrum. By way of an example, we consider how such problems arise, what comprises their mathematical content, and what conclusions can be drawn from results obtained. We will present an estimate of the length of codes for signals with finite spectrum and discuss it in connection with the problems of sound recording.

Raising of the question.
Of the sound recording techniques the most widely used method is the so-called analogue method. When using this method a signal to be retained is recorded in its natural form without any preceding transformations. This system of recording is remarkable for its simplicity. The system's disadvantage is the impossibility of defending signals from interference. All defects of recording and reproducing devices, the inhomogeneity and ageing of materials and the like lead to distortions in reproducing.

Another method of recording in which we are interested, the digital one, consists of the following: The signal is transformed into a discrete code, the code of the signal is recorded and, in order to be reproduced, it is again transformed into its natural continuous form. As far as this system is concerned, there are many ways of protecting signals from various sorts of noises. But in sound reproduction this system is not used because the existing schemes of coding still remain unacceptably complex.

Successful development of a digital recording system requires the construction of a handy mathematical model of sound signals and the discovery of simple schemes of coding. The first question arising is: How long must the codes be for the signals to be reproduced with a desirable accuracy? The qualitative estimate obtained seems encouraging.
The trip to Canada for the International Congress of Mathematicians was not Vitushkin's only trip to North America. He delivered a series of lectures at the University of California, Los Angeles, USA, in April-May 1977 which were published under the title On representation of functions by means of superpositions and related topics.

Finally, we give a version of the tribute [5] written following Vitushkin's sudden death at the age of 72:-
On 9 May 2004, at the age of 72, the outstanding mathematician, the acknowledged leader in complex analysis, member of the Editorial Board of our journal [Matematicheskie Zametki], Full Member of the Russian Academy of Sciences Anatolii Georgievich Vitushkin died suddenly and unexpectedly.

To A G Vitushkin we owe deep and fundamental results in various branches of contemporary real and complex analysis, approximation theory, and complex geometry. He obtained theorems on the impossibility of expressing smooth functions of several variables in terms of superpositions of smooth functions of a smaller number of variables, which is one of the most prominent achievements toward the solution of the 13th Hilbert problem, while his estimates of the complexity of algorithms for calculating and approximating functions of various classes have been incorporated into textbooks on computational methods and mathematical modelling. The Vitushkin criterion for the uniform approximation of holomorphic functions by rational functions has rounded off a century of the development of classical approximation theory in the complex domain and initiated a new period of its development. Vitushkin's results concerning the geometry of real hypersurfaces of multidimensional complex space and the theory of holomorphic mappings between them have led to the creation of new approaches and the development of new methods in multidimensional complex analysis.

A G Vitushkin was the founder and head of a leading scientific school in complex analysis and approximation theory. Fundamental research in multidimensional complex analysis in our country is inseparably linked with his name. A G Vitushkin's deep original ideas and his inspiring personality have exerted influence on practically all scientists who have been working in complex analysis during the past forty years. In addition to numerous courses of lectures brilliantly delivered at the Mechanics and Mathematics Department of Moscow University, his extensive and many-sided pedagogical activity encompassed his well-known research seminar on complex analysis and the creation of a remarkable system of math circles and seminars for junior students.

A G Vitushkin's remarkable research results and the scientific school created by him have played, and will continue to play, a crucial role in the development of complex analysis and geometry in the whole world.

The bright image of Anatolii Georgievich Vitushkin will remain in our hearts for ever.

### References (show)

1. A A Bolibruch, Yu S Osipov and Ya G Sinai (eds.), Mathematical events of the Twentieth Century (Springer, 2006).
2. Anatoli Georgievich Vitushkin, Mathematical Institute, Russian Academy of Sciences. http://www.mi-ras.ru/index.php?c=inmemoriapage&id=11659
3. V K Beloshapka, Anatolii Georgievich Vitushkin (on his 70th birthday), Russian Mathematical Surveys 57 (2002), 183-190.
4. V K Beloshapka, Anatolii Georgievich Vitushkin (on his 70th birthday) (Russian), Uspekhi Mat. Nauk 57 (1) (2002), 179-184.
5. Editors, Anatoli Georgievich Vitushkin, Mathematical Notes 76 (3) (2004), 305.
6. A G Vitushkin, Half a Century as One Day, in A A Bolibruch, Yu S Osipov and Ya G Sinai (eds.), Mathematical events of the Twentieth Century (Springer, 2006), 475-529.
7. L C Young, Review: On multidimensional variations, by A G Vitushkin, Mathematical reviews MR0075267 (17,718f).