2019
Том 71
№ 9

# Chaichenko S. O.

Articles: 10
Article (Ukrainian)

### Approximation of bounded holomorphic and harmonic functions by Fejér means

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 516-542

We compute the exact values of the exact upper bounds on the classes of bounded holomorphic and harmonic functions in a unit disk for the remainders in a Voronovskaya-type formula in the case of approximation by Fej´er means. We also present some consequences that are of independent interest.

Article (Ukrainian)

### Approximation of the Bergman kernels by rational functions with fixed poles

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1577-1584

We solve the problem of best rational approximations of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.

Article (Ukrainian)

### Best Approximations for the Cauchy Kernel on the Real Axis

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1540–1549

We compute the values of the best approximations for the Cauchy kernel on the real axis $ℝ$ by some subspaces from $L_q (ℝ)$. This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka–Malmquist system from the functions lying in the unit ball of the Hardy space $H_p,\; 2 ≤ p < ∞$.

Article (Russian)

### Approximations by Fourier Sums on the Sets $L^{ψ} L^{P(∙)}$

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 835–846

We study some problems of imbedding of the sets of $ψ$-integrals of the functions $f \in L^{p(∙)}$ and determine the orders of approximations of functions from these sets by Fourier’s sums.

Article (Russian)

### Best approximations of periodic functions in generalized lebesgue spaces

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1249-1265

In generalized Lebesgue spaces with variable exponent, we determine the order of the best approximation on the classes of $(\psi, \beta)$-differentiable $2\pi$-periodic functions. We also obtain an analog of the well-known Bernstein inequality for the $(\psi, \beta)$-derivative, with the help of which the converse theorems of approximation theory are proved on the indicated classes.

Article (Ukrainian)

### Approximation of classes of analytic functions by a linear method of special form

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 102-109

On classes of convolutions of analytic functions in uniform and integral metrics, we find asymptotic equations for the least upper bounds of deviations of trigonometric polynomials generated by certain linear approximation method of a special form.

Article (Ukrainian)

### Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 968–978

For the least upper bounds of deviations of the de la Vallée-Poussin operators on the classes $\widehat{L}^{\psi}_{\beta}$ of rapidly vanishing functions $ψ$ in the metric of the spaces $\widehat{L}_p,\; 1 ≤ p ≤ ∞$, we establish upper estimates that are exact on some subsets of functions from $\widehat{L}_p$.

Article (Ukrainian)

### Approximation of Analytic Periodic Functions by de la Vallée-Poussin Sums

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1653-1669

We investigate the approximation properties of the de la Vallée-Poussin sums on the classes $C_{\beta }^q H_{\omega }$ . We obtain asymptotic equalities that, in certain cases, guarantee the solvability of the Kolmogorov–Nikol'skii problem for the de la Vallée-Poussin sums on the classes $C_{\beta }^q H_{\omega }$ .

Article (Russian)

### Approximation of the Classes $C^{{\bar \psi }} H_{\omega }$ by de la Vallée-Poussin Sums

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 681-691

We investigate the problem of the approximation of the classes $C^{{\bar \psi }} H_{\omega }$ introduced by Stepanets in 1996 by the de la Valée-Poussin sums. We obtain asymptotic equalities that give a solution of the Kolmogorov–Nikol'skii problem for the de la Valée-Poussin sums on the classes Cψ¯HωCψ¯Hω in several important cases.

Article (Russian)

### Approximation of $\overline \psi$-Integrals of Periodic Functions by de la Vallée-Poussin Sums (Low Smoothness)

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1641-1653

We investigate the asymptotic behavior of the upper bounds of deviations of linear means of Fourier series from the classes $C_{\infty} ^{\psi}$. In particular, we obtain asymptotic equalities that give a solution of the Kolmogorov – Nikol'skii problem for the de la Vallée-Poussin sums on the classes $C_{\infty} ^{\psi}$.