Uniform Approximations by the Poisson Threeharmonic Integrals on the Sobolev Classes

Abstract

The quantity of the precise upper bound of the deviations of the linear methods of summation, which are determined by rectangular number matrix Λ = IIλn,kII on the classes of continuous periodic functions in the uniform metric is given. As possible application of the obtained results, we study the asymptotic behavior of threeharmonic Poisson integrals in the case when the classes W r∞, r ∈ N , are an object of approximation. The asymptotic equalities reveal the theoretical foundations and mathematical features of one of the main problems of approximation theory − the Kolmogorov-Nikol'sky problem. In particular, the problem is solved for the threeharmonic Poisson integrals on the Sobolev classes in the uniform metric. It is found that the threeharmonic Poisson integrals possess approximation properties that are different from the properties of the harmonic and biharmonic Poisson integrals, which were studied previously, and some concepts and techniques of approximation theory can also be useful in studying the spaces of functions with generalized derivatives. An important moment in the solution of this problem is the fact that with the help of the asymptotic equalities, which are studied, a wide range of economic problems can be solved, the solution of which by methods of classical linear algebra and mathematical analysis is a complicated process. Economic modeling and forecasting on the basis of the constructed mathematical model can be used in the analysis of processes of economic dynamics, considering polyharmonic regimes. The purpose of the work is to develop a mathematical apparatus that allows to build mathematical models of periodic economic processes. Modeling serves as a means of analyzing the economy and the phenomena occurring in it, as well as justifying the decisions made, forecasting and managing economic processes and objects. We also analyze some fundamental problem of the modern economy, solved by methods of the approximation theory