Estimate of the Rate of Approximation by Images of Operators of Abel-Poisson Type of Some Special Classes of Functions

Abstract

The properties of Abel−Poisson type operators, which have a wide range of applications in various fields of scientific research are studied. Special attention is paid to approximation and differential properties of operators of Abel−Poisson type. In particular, an estimate was obtained for the rate of approximation by the images of operators of Abel−Poisson type for functions having a given majorant of the type of the second modulus of continuity of the r-th derivative by the Vallée-Poussin sum of (n, 2n) order in the integral metric Lp(D). For further consideration of differential properties of operators of Abel−Poisson type, we present the definitions of classes of functions that are a generalization of classes of differentiable functions of S.M. Nikolskiy. Operators of the Abel−Poisson type are one of the main that are used in real and complex analysis and mathematical physics, and their images, which are differentiable functions, are often accepted as the solutions of known boundary value problems. Therefore, the result obtained in the paper can be used to study the boundary properties of operators of Abel−Poisson type and is adjoint to similar results of Ya.S. Bugrov. In addition, it is possible to use the described properties of these operators in the theory of dynamic game problems, which is especially important nowadays, for example, to find stationary targets that have crashed and are in practically inaccessible places, when developing computer search systems and monitoring moving objects, in the analysis and modeling of group interaction between moving objects. Such tasks often arise in the maintenance of sea and air transport.