The paper is devoted to a behavior investigation of the upper bound of deviation of functions from Zygmund classes from their biharmonic Poisson integrals. Systematic research in this direction was conducted by a number of Ukrainian as well as foreign scientists. But most of the known results relate to an estimation of deviations of functions from different classes from operators that were constructed based on triangular l-methods of the Fourier series summation (Fejer, Valle Poussin, Riesz, Rogozinsky, Steklov, Favard, etc.). Concerning the results relating to linear methods of the Fourier series summation, given by a set of functions of natural argument (Abel-Poisson, Gauss-Weierstrass, biharmonic and threeharmonic Poisson integrals), in this direction the progress was less notable. This may be due to the fact that the above-mentioned linear methods the Fourier series summation are solutions of corresponding integral and differential equations of elliptic type. And, therefore, they require more time-consuming calculations in order to obtain some estimates, that are suitable for a direct use for applied purposes. At the same time, in the present paper we investigate approximative characteristics of linear positive Poisson-type operators on Zygmund classes of functions. According to the well-known results by P.P. Korovkin, these positive linear operators realize the best asymptotic approximation of functions from Zygmund classes. Thus, the estimate obtained in this paper for the deviation of functions from Zygmund classes from their biharmonic Poisson integrals (the least studied and most valuable among all linear positive operators) is relevant from the viewpoint of applied mathematics.

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.