ON APPROXIMATION OF FUNCTIONS FROM ZYGMUND CLASSES BY BIHARMONIC POISSON INTEGRALS

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.

EXACT EQUALITIES FOR APPROXIMATION OF FUNCTIONS FROM THE SOBOLEV CLASS BY THEIR GENERALIZED POISSON INTEGRALS

In most cases, solutions to problems of the motion of a system of interacting material points are reduced to either ordinary differential equations or partial differential equations. One of the solutions of this type of equations is the so-called generalized Poisson integrals, which in partial cases turn into the well-known Abel-Poisson integrals or biharmonic Poisson integrals. A number of results is known on the approximation of various classes of differentiable periodic and nonperiodic functions by the mentioned above integrals (the so-called Kolmogorov-Nikol’skii problem in the terminology of A.I. Stepanets). Nevertheless, there is a significant drawback practically in all of the solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals from the mathematical modeling (computational experiment) point of view. The core point here is that in most of the previously solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals only the leading and remainder terms of the approximation were obtained, that can significantly affect the accuracy of the computational experiment. In the present paper we obtain exact equalities for approximation of functions from the Sobolev classes by their generalized Poisson integrals. Consequently, the theorem proved in this paper is a generalization and refinement of previously known results characterizing the approximation properties of Abel-Poisson integrals and biharmonic Poisson integrals on the classes of differentiable periodic functions. A peculiarity of the solved in this work problem of approximation for the generalized Poisson integral on the classes of differentiable functions is that the result obtained is successfully written using the well-known Akhiezer-Krein-Favard constants. This fact substantially increases the accuracy of the mathematical modeling result (computational experiment) for a real process described using the generalized Poisson integral. These results can further significantly expand the scope of application of the Kolmogorov-Nikol’skii problems to mathematical modeling.

Approximative properties of the three-harmonic Poisson integrals on the classes $W^{r}_{\beta}H^{\alpha}$.

We obtained the asymptotic equalities for the least upper bounds of the approximation of functions from the classes $W^{r}_{\beta}H^{\alpha}$ by three-harmonic Poisson integrals in the case $r+\alpha\leq 3$ in the uniform metric.

Asymptotics of approximation of functions by conjugate Poisson integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.