S e r g e i L' v o v i c h S o b o l e v

Key words: Institute of Mathematics of the Siberian Branch,
Siberian Branch of the Academy of Sciences, theory of cubature formulas

In Siberia Sergei L'vovich at the building site of the Institute of Mathematics at Academgorodok

The Siberian period of S.L.Sobolev's scientific work was marked by substantial achievements in the theory of cubature formulas. The problem of approximate integration of functions is one of the main problems in the theory of computations; computation for this problem is extremely laborious in regard to multidimensional integrals. To find the best cubature formula and obtain the exact estimate of its error is a very difficult problem for classes of functions encountered in applications. It requires to draw deep and subtle results from other fields of mathematics. The problem of optimizing the integration formulas in a modern understanding becomes the problem of finding a minimum of the norm of the error l which is given on some space of functions.
Sergei L'vovich developed the theory of cubature formulas for a sphere which are invariant under some group of rotations G. He proved the fundamental theorem that invariant cubatures which integrate exactly all invariant G spherical harmonics of some order will also integrate exactly all other harmonics of this order. Assuming that the cubature nodes are placed at the vertices of some regular frame in the n-dimensional space with the step h, and the weights are subject to the condition of the cubature exactness on all polynomials of degree m>n/2, S.L.Sobolev considered the cubatures of composite type which are obtained by summation of cubatures for small domains. He established that for such cubatures to within the terms of higher order of smallness the equality ||l||=Ah^m holds where the value A is represented in terms of the Epstein function which depends on the shape of the grid. S.L.Sobolev introduced and studied cubature formulas of special form, the so-called formulas with regular boundary layer which generalize the well-known Gregory cubature formulas to functions of many variables. In those formulas, the weights are equal outside the boundary layer, and the formulas themselves possess the asymptotically minimal norm of the error. Sobolev's investigations in cubature formulas were continued by a large group of his students.

S.L.Sobolev's research work was inseparable from his management in science. At the end of the 50s Academicians M.A.Lavrent'ev, S.L.Sobolev, and S.A.Khristianovich came out with the initiative to organize a new big scientific center, the Siberian Branch of the Academy of Sciences. For many scientists of the SB AS of the first enrolment the example of Sergei L'vovich, the attractiveness of his personality, and his scientific authority were a forcible argument in making a solution to move to Novosibirsk for work.
The role of Sergei L'vovich in formation of the Siberian mathematical
school cannot be overestimated. The founder of the Institute of Mathematics of the Siberian Branch and its director in the course of a quarter of century, S.L.Sobolev made a decisive contribution to determination of the scientific destiny of the Institute which now bears his name. Also large is the role of S.L.Sobolev in formation of the Novosibirsk State University where he founded the Chair of Differential Equations. S.L.Sobolev worked actively in the Bureau of the Department of Mathematics, was a member of the Presidium of the Siberian Branch from the moment of its foundation, was the Chairman of the National Committee of Mathematicians, was elected many times a deputy of councils of various levels.
Academician S.L.Sobolev was a member of the Editorial Board of the Siberian Mathematical Journal from the moment of its foundation. He was the Editor-in-Chief of the Journal from 1968 till 1988. The reputation and aspect of the Journal in those decades were in many points determined by Sobolev's scientific and ethical principles and personal qualities.

S.L.Sobolev's scientific and organizational deserts were highly appraised by the State government: he was decorated with many orders and medals. S.L.Sobolev's scientific work gained world-wide acknowledgement. He was an honorary doctor of Humbold University in Berlin, an honorary doctor of Karlow University in Prague, an honorary doctor of the Higher School of Architecture and Construction in Weimar. S.L.Sobolev was a foreign member of the French Academy of Sciences, a foreign member of the National Academy dei Lincei in Rome, a foreign member of the GDR Academy of Sciences in Berlin, an honorary member of the Edinburgh Royal Society, an honorary member of the Moscow Mathematical Society and American Mathematical Society. In 1978 S.L.Sobolev was decorated by the Czecho-Slovak Academy of Sciences with the Gold Medal "For Services to Science and Humanity" and in 1981 with the Bolzano Gold Medal. In 1987 he was decorated with the Silver Medal of the Czecho-Slovak Academy of Sciences. In 1989 S.L.Sobolev was awarded the highest prize of the Russian Academy of Sciences, the Lomonosov Gold Medal.

The versatility of Sobolev's talents was manifested in his passion for music, literature, and poetry. His great erudition, original ideas, his skill to find barely noticeable but essential details in issues under discussion, rigoruos logic, represible humour, and human charm made him a brilliant interlocutor and polemist. Sergei L'vovich was distinguished by extraordinary generosity of soul, optimism, goodwill and confidence in people, deepness and clearness of mind, modesty, and sympathy.

S.L.Sobolev's scientific ideas came into the treasure-trove of world science and became a property of many generations of present day and future mathematicians. They are destined to adorn our science for long time.

M.M. Lavrent'ev, V.L. Beresnev,
A.A. Borovkov, S.K. Godunov,
Yu.L. Ershov, S.S. Kutateladze,
Yu.G.Reshetnyak, V.G. Romanov.

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