Riassunto
The fates of important mathematical ideas are varied. Sometimes they are instantly appreciated by the specialists and constitute the foundation of the development of theories or methods. It also happens, however, that even ideas uttered by distinguished mathematicians are surrounded with respectful indifference for a long time, and every effort of inter preters and successors has to be made in order to gain for them the merit deserved. It is the second case that is encountered in the present book, the author of which, the Leningrad mathematician E.M. Polishchuk, reconstructs and develops one of the dir.ctions in functional analysis that originated from Hadamard and Gateaux and was newly thought over and taken as the basis of a prospective theory by Paul Levy. Paul Levy, Member of the French Academy of Sciences, whose centenary of his birthday was celebrated in 1986, was one of the most original mathe matiCians of the second half of the 20th century. He could not complain about a lack of attention to his ideas and results. Together with A.N. Kolmogorov, A.Ya. Khinchin and William Feller, he is indeed one of the acknowledged founders of the theory of random processes. In the proba bility theory and, to a lesser degree, in functional analysis his work is well-known for its conceptualization and scope of the problems posed.
Contenuti
1. Functional Classes and Function Domains. Mean Values. Harmonicity and the Laplace Operator in Function Spaces.- 1. Functional classes.- 2. Function domains.- 2.1. Uniform domains.- 2.2. Normal domains.- 3. Continual means.- 3.1. The mean over a uniform domain.- 3.2. The mean value ma, RF over the Hilbert sphere and its main properties.- 3.3. The spherical mean of a Gâteaux functional.- 3.4. Functionals as rarndom variables.- 3.5. The Dirac measure in a function space. The centre of a function domain. Harmonicity.- 4. The functional Laplace operator.- 4.1. Definitions and properties.- 4.2. Spherical means and the Laplace operator in the Hilbert co-ordinate space 12.- 2 Chapter 2. The Laplace and Poisson Equations For a Normal Domain.- 5. Boundary value problems for a normal domain with boundary values on the Gâteaux ring.- 5.1. Functional Laplace and Poisson equations.- 5.2. The fundamental, functional of a surface S.- 5.3. Examples.- 5.4. The maximum principle and uniqueness of solutions.- 5.5. The exterior Dirichlet problem.- 5.6. The deviation H – F.- 6. Semigroups of continual means. Relations to the probability solutions of classical boundary value problems. Applications of the integral over a regular measure.- 6.1. Semigroups of means over Hilbert spheres.- 6.2. The operators ms, n and the probability solutions of classical boundary value problems in the space Em.- 6.3. Regular measures and the extension of the Gâteaux ring.- 3. The Functional Laplace Operator and Classical Diffusion Equations. Boundary Value Problems for Uniform Domains. Harmonic Controlled Systems.- 7. Boundary value problems with strong Laplacian and their parallelism to classical parabolic equations.- 7.1. The functional Laplacian and the classical parabolic operator.- 7.2. Dual problems and an analogy table.- 8. Boundary value problems for uniform domains.- 8.1. Functional and classical Dirichlet problems.- 8.2. The Dirichlet problem for operators.- 8.3. The functional Neumann problem.- 8.4. Properties of the Poisson equation.- 9. Harmonic control systems.- 9.1. Normal control domain.- 9.2. Uniform control domain.- 4. General Elliptic Functional Operators on Functional Rings.- 10. The Dirichlet problem in the space of summable functions and related topics.- 10.1. Functional elliptic operators of general type.- 10.2. Compact extensions of function domains. Compact restrictions.- 10.3. Averaging M?,?F of a functional $$[F \in \overline R$$ with respect to a family of transition densities of diffusion processes.- 11. The generalized functional Poisson equation.- Comments.- References.
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