On Some Applications of Isometricity of Functional Spaces in Applied Mathematics

Abstract

Problems of computational mathematics are directly connected with an implementation of mathematical models in conditions of limited initial information. This is especially evident when, during modeling, one encounters various idealizations of real processes which in turn forces one to apply discretization for functions of continuous variables, as well as replacing all infinitesimal and infinitely large quantities with some finite quantities. That is why the search for a mathematical description of the model or the choice between several possible ones is the most difficult and crucial moment in modeling, since the model may contain a sufficiently large number of connections, parts, variables and choosing the wrong mathematical description for any of them can yield a full or partial operability of the model as a whole. To describe the interactions one selects a priory known functional dependencies. One of the most interesting problems of computational mathematics that require a delicate approach to choosing both the space in which the problem is solved and its capability to scale up quickly the results from a small number of parameters to a much larger number of them, are exactly the problems of approach-evasion differential games, the problems of escape or evasion from a meeting, the problems of escape from a group of pursuers. In solving these problems and in applying the results to minimize redundancy of calculations, it is certainly important to select appropriately a functional space with possible inherent isometry properties. The choice of such space is a separate task requiring a deep comprehensive study. The spaces of real functions of n + k variables that are isometric to the spaces of real functions defined on the n-dimensional Euclidean space are already studied in detail. In this paper, isometric mappings are considered of the spaces of real functions of n + m variables into the spaces of real functions of n variables that are 2π-periodic in each variable, which in turn will contribute to the study of complex controlled systems, as well as finding an optimal mathematical models for such systems.