Approximation of the classes $W^{r}_{\beta,\infty}$ by three-harmonic Poisson integrals

Abstract

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes

“…From ( 27), taking into account ( 28) and ( 29 Similarly to [27] we can show that the following equality holds Thence…”

Approximation of Classes $$ {C}_{\beta, \infty}^{\psi } $$ by Three-Harmonic Poisson Integrals in Uniform Metric (Low Smoothness)

“…It is known (see, e.g., [7,11]) that the values of approximation by the Poisson and biharmonic Poisson integrals cannot tend to zero at δ → ∞ faster than 1 δ and 1 δ 2 , respectively. At the same time, as follows from works [17][18][19], using the approximation by three-harmonic integrals, one can obtain the approximation rate 1 δ 3 , δ → ∞. Therefore, our purpose is to study the asymptotic behavior of the quantities E(C ψ β,∞ ; P 3,δ ) C at δ → ∞.…”

“…The first results related to the study of the approximative properties of three-harmonic Poisson integrals were obtained in work [6]. Later, the research in this direction was continued in works [17][18][19]. In particular, the Kolmogorov-Nikolskii problem for the three-harmonic Poisson integral on the classes W r β,∞ , r > 0, β ∈ R, was solved in [17], and on the classes [18,19].…”

“…Later, the research in this direction was continued in works [17][18][19]. In particular, the Kolmogorov-Nikolskii problem for the three-harmonic Poisson integral on the classes W r β,∞ , r > 0, β ∈ R, was solved in [17], and on the classes [18,19]. As concerning the study of the approximative properties of three-harmonic Poisson integrals on the classes of (ψ, β)-differentiable functions, this issue has not been studied enough.…”

“…where A(τ ) is defined by equality (2.2), τ (u) is given by relationship (2.1), and its transformation τ β (t) is summable on the whole numerical axis. Using the methods for studying the residual term on the right-hand side of equality (2.38) (they were presented in work [17]), we obtain the following estimate:…”