Isometry of the Functional Spaces with Different Number of Variables and Some its Applications in the Theory of Approximation of Functions
Year: 2019
Type: article
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 we have or can compute only some points in which values of the function are known, and, in addition, these data are 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. The point is that it is desirable to simplify the mathematical model of a complex system as much as possible. 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 functions to be approximated, it is necessary to choose from it one definite function by means of some criterion. In the paper, using isometric mappings of the spaces of real functions of 1+m variables to the spaces of univariate 2π -periodic functions from one variable, we study the approximate characteristics of 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.
Author: Dmitriy N. Bushev
Institution: Lesya Ukrainka Volyn National University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2943877397