# Sokolenko I. V.

### Approximation by interpolation trigonometric polynomials in metrics of the spaces $L_p$ on the classes of periodic entire functions

Serdyuk A. S., Sokolenko I. V.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 283-292

We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with equidistant distribution of interpolation nodes $x_{(n 1)}^k = \frac{2k\pi}{2n 1}, k \in Z,$, in metrics of the spaces $L_p$ on the classes of $2\pi$ -periodic functions that can be represented in the form of convolutions of functions $\varphi , \varphi \bot 1$, from the unit ball of the space $L_1$, with fixed generating kernels in the case where the modules of their Fourier coefficients $\psi (k)$ satisfy the condition $\mathrm{lim}_{k\rightarrow \infty} \psi (k + 1)/\psi (k) = 0.$. Similar estimates are also obtained on the classes of $r$-differentiable functions $W^r_1$ for the rapidly increasing exponents of smoothness $r (r/n \rightarrow \infty , n \rightarrow \infty )$.

### Oleksandr Ivanovych Stepanets’ (on his 75th birthday)

Romanyuk A. S., Romanyuk V. S., Samoilenko A. M., Savchuk V. V., Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 579

### International conference "Theory of approximation of functions and its applications" dedicated to the 70 th birthday of the corresponding member of NASU Professor O. I. Stepanets (1942 - 2007)

Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1438-1440

### Linear approximation methods and the best approximations of the Poisson integrals of functions from the classes $H_{ω_p}$ in the metrics of the spaces $L_p$

Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 979–996

We obtain upper estimates for the least upper bounds of approximations of the classes of Poisson integrals of functions from $H_{ω_p}$ for $1 ≤ p < ∞$ by a certain linear method $U_n^{*}$ in the metric of the space $L_p$. It is shown that the obtained estimates are asymptotically exact for $р = 1$: In addition, we determine the asymptotic equalities for the best approximations of the classes of Poisson integrals of functions from $H_{ω_1}$ in the metric of the space $L_1$ and show that, for these classes, the method $U_n^{*}$ is the best polynomial approximation method in a sense of strong asymptotic behavior.

### International Conference "Mathematical Analysis and Differential Equations and Applications"

Samoilenko A. M., Savchuk V. V., Sokolenko I. V., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 431

### Approximation of the $\bar {\Psi}$ -integrals of functions defined on the real axis by Fourier operators

Sokolenko I. V., Stepanets O. I.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 960–965

We find asymptotic formulas for the least upper bounds of the deviations of Fourier operators on classes of functions locally summable on the entire real axis and defined by $\bar {\Psi}$-integrals. On these classes, we also obtain asymptotic equalities for the upper bounds of functionals that characterize the simultaneous approximation of several functions.

### Approximation of $\bar {\omega}$ -integrals of continuous functions defined on the real axis by Fourier operators

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 663-676

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functions $C^{ψ}_{∞}$ and $\hat{C}^{\bar{\psi} } H_{\omega}$ in the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of $\bar{\psi}$-integrals of continuous functions by Fourier operators in the uniform metric.