Isometry of the Functional Spaces with Different Number of Variables and Some its Applications in the Theory of Approximation of Functions

Abstract

Among the most important scientific problems of computational mathematics are those, that are connected with the implementation of mathematical models in conditions of limited initial information, when all that can be calculated are some points with known values of the function, and mostly approximate due to errors of different origin. Theoretical study of complex controlled systems, finding the optimal mathematical models of such systems requires research in the field of building controllability criteria. In general, creating a good mathematical model is an art. It is desirable to simplify as much as possible the mathematical model of a complex system. At the same time, with such simplification, the description of those features of the behavior to be investigated should not disappear. The main criterion here is the correspondence of the mathematical model to the described real processes. It is determined by a comparison of the results of the theoretical calculation with the results of the experiment at a particular object. The model deserves special recognition if it helps to reveal theoretically new behavioral features, which are confirmed experimentally. In this connection, it is extremely important to study in detail the features of the space in which the model is constructed, the norms in this space, and the construction of approximations for functions in this space. The classical approach in the approximation theory consists in using the available information to obtain the approximated function, which is fairly easy to operate. Having determined the class of approximated functions, one needs to choose from it one concrete by using some criterion. In the paper, using isometric mappings of the spaces of real functions of 1+m variables on the spaces of 2π-periodic functions from one variable, one has studied the approximation characteristics of the classes of functions in these spaces. The results obtained open up many possibilities for further research in the asymptotic theory of representations. The described mathematical models can be used in solving problems of computational mathematics related to the study of complex controlled systems.