Continual Means and Boundary Value Problems in Function Spaces 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. 1. Functional Classes and Function Domains. Mean Values. Harmonicity and the Laplace Operator in Function Spaces 1. Functional classes 2. Function domains 3. Continual means 4. The functional Laplace operator 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 6. Semigroups of continual means. Relations to the probability solutions of classical boundary value problems. Applications of the integral over a regular measure 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 8. Boundary value problems for uniform domains 9. Harmonic control systems 4. General Elliptic Functional Operators on Functional Rings 10. The Dirichlet problem in the space of summable functions and related topics 11. The generalized functional Poisson equation Comments References.