# Generalized Functions, Integral Geometry and Representation Theory

Brand new Book. The first systematic theory of generalized functions also known as distributions was created in the early s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics.

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Graev, N. New Hardcover Quantity Available: 1. The concept of a generalized or weak solution of a differential equation was considered earlier. However, it was exactly in the works by Sobolev that this concept was elaborated and applied systematically. Sobolev posed and solved the Cauchy problem in spaces of functionals, which was based on the revolutionary extension of the Eulerian concept of function and declared as the date of the birth of the theory of distributions. Suggesting his definition of generalized derivative, Sobolev enriched mathematics with the spaces of functions whose weak derivatives are integrable to some power.

These are now called Sobolev spaces. Sobolev found the general criteria for equivalence of various norms on W p l and showed that these spaces are the natural environment for posing the boundary value problems for elliptic equations. This conclusion was based on his thorough study of the properties of Sobolev spaces.

The most important facts are embedding theorems. Each embedding theorem estimates the operator norm of an embedding, yielding special inequalities between the norms of one and the same function inside various spaces. In Sobolev was elected a corresponding member of the Academy of Sciences at the age of 24 years. In he became a full member of the Academy and remained the youngest academician for many years. Inspired by military applications in the s, Sobolev studying the system of differential equations describing small oscillations of a rotating fluid. He obtained the conditions for stability of a rotating body with a filled-in cavity which depend on the shape and parameters of the cavity.

This research by Sobolev signposted another area of the general theory which concerns the Cauchy and boundary value problems for the equations and systems that are not solved with respect to higher time derivatives. In the grievous years of the Second World War from to Sobolev occupied the position of the director of the Steklov Mathematical Institute. Sobolev was one of the first scientists who foresaw the future of computational mathematics and cybernetics.

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From to he held the chair of the first national department of computational mathematics at Moscow State University. This department has played a key role in the development of this important area of the today's mathematics. Addressing the problems of computational mathematics, Sobolev lavishly applied the apparatus of the modern sections of the theoretical core of mathematics.

It is typical for him to pose the problems of computational mathematics within functional analysis. It is worthwhile to emphasize the great role in the uprise of cybernetics, genetics, and other new areas of research in this country which was played by the publications and speeches of Sobolev who valiantly defended the new trends in science from the ideologized obscurantism.

It is difficult to overrate the contribution of Sobolev to the design of the nuclear shield of this country. The main task of the joint work with I. Sobolev administered and supervised various computational teams, studied the control of the industrial processes of isotope separation, struggled for the low costs of production and made decisions on many managerial and technological matters. Sobolev's research was inseparable from his management in science. At the end of the s M. Sobolev, and S. Khristianovich came out with the initiative to organize a new big scientific center, the Siberian Division of the Academy of Sciences.

For many scientists of the first enrolment to the Siberian Division it was the example of Sobolev, his authority in science, and the attraction of his personality that yielded the final argument in deciding to move to Novosibirsk. The Siberian period of Sobolev's life in science was marked by the great achievements in the theory of cubature formulas.

Approximate integration is one of the main problems in the theory of computations-the cost of computation of multidimensional integrals is extremely high. Optimizing the formulas of integration is understood now to be the problem of minimizing the norm of the error on some function space. Sobolev suggested new approaches to the problem and discovered marvelous classes of optimal cubature formulas.

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Sobolev merits brought him many decorations and signs of distinction. There were quite a few prominent persons among his next of kin. Hadamard was his granduncle. It is curious that Hadamard had a low opinion of Schwartz mathematical plans, since the sixteen-years old Laurent did not know the Riemann zeta function. Cartan, A. Denjoy, M. Lebesgue who delivered lectures and Hadamard who conducted seminars. After graduation from the ENS Schwartz decided to be drafted in the compulsory military service for two years. He had to stay in the army in in view of the war times.

These years were especially hard for the young couple of the Schwartzes. It was unreasonable for Jews to stay in the occupied zone.

## Series: Generalized Functions

The Schwartzes had to escape from the native north and manage to survive on some modest financial support that was offered in particular by Michelin, a world-renowned tire company. In Schwartz was in Toulouse for a short time and met H. Cartan and J. Delsarte who suggested that the young couple should move to Clermont-Ferrand, the place of temporary residence of the group of professors of Strasbourg University which had migrated from the German occupation. These were J.

Ehresmann, A.

## Integral geometry - Encyclopedia of Mathematics

Lichnerowicz, and S. In Clermont-Ferrand Schwartz completed his Ph. Thesis on approximation of a real function on the axis by sums of exponentials. Unfortunately, the war had intervened into the mathematical fate of Schwartz. His family had to change places under false identities.