# Shydlich A. L.

### Direct and inverse theorems on the approximation of 2π -periodic functions by Taylor – Abel – Poisson operators

Prestin J., Savchuk V. V., Shydlich A. L.

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 657

We obtain direct and inverse theorems on the approximation of 2\pi -periodic functions by Taylor – Abel – Poisson operators in the integral metric.

### Order equalities for some functionals and their application to the estimation of the best $n$-term approximations and widths

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1403-1423

We study the behavior of functionals of the form $\sup_{l>n} (l-n)\left(∑^l_{k=1} \frac1{ψ^r(k)} \right)^{−1/r}$, where $ψ$ is a positive function, as $n → ∞$: The obtained results are used to establish the exact order equalities (as $n → ∞$) for the best $n$-term approximations of $q$-ellipsoids in metrics of the spaces $S^p_{φ}$: We also consider the applications of the obtained results to the determination of the exact orders of the Kolmogorov widths of octahedra in the Hilbert space.

### Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis

Doronin V. G., Ligun A. A., Serdyuk A. S., Shydlich A. L.

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 92-98

We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.

### Classification of infinitely differentiable periodic functions

Serdyuk A. S., Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1686–1708

The set $\mathcal{D}^{\infty}$ of infinitely differentiable periodic functions is studied in terms of generalized
$\overline{\psi}$-derivatives defined by a pair $\overline{\psi} = (\psi_1, \psi_2)$ of sequences $\psi_1$ and $\psi_2$.
In particular, it is established that every function $f$ from the set $\mathcal{D}^{\infty}$ has at least one derivative whose parameters $\psi_1$ and $\psi_2$
decrease faster than any power function. At the same time, for an arbitrary function $f \in \mathcal{D}^{\infty}$ different from
a trigonometric polynomial, there exists a pair $\psi$ whose parameters $\psi_1$ and $\psi_2$ have the same rate of decrease
and for which the $\overline{\psi}$-derivative no longer exists.

We also obtain new criteria for $2 \pi$-periodic functions real-valued on the real axis to belong to the set of
functions analytic on the axis and to the set of entire functions.

### Saturation of the linear methods of summation of Fourier series in the spaces *S*^{ p}_{φ}

^{ p}

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 815 – 828

We consider the problem of the saturation, in the spaces *S ^{ p}*

_{φ}, of linear summation methods for Fourier series, which are determined by the sequences of functions defined on a subset of the space

*C*. We obtain sufficient conditions for the saturation of such methods in these spaces.

### On some new criteria for infinite differentiability of periodic functions

Serdyuk A. S., Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1399–1409

The set $\mathcal{D}^{\infty}$ of infinitely differentiable periodic functions is studied in terms of generalized $\overline{\psi}$-derivatives defined by a pair $\overline{\psi} = (\psi_1, \psi_2)$ of
sequences $\psi_1$ and $\psi_2$ .
It is shown that every function $f$ from the set $\mathcal{D}^{\infty}$ has at least one derivative whose parameters $\psi_1$ and $\psi_2$ decrease faster than any power function, and, at the same time, for an arbitrary
function $f \in \mathcal{D}^{\infty}$ different from a trigonometric polynomial, there exists a pair $\psi$ whose parameters $\psi_1$ and $\psi_2$ have the same rate of decrease and for which the $\overline{\psi}$-derivative no longer exists.

### On some properties of convex functions

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 920–938

We obtain some new results for convex-downward functions vanishing at infinity.

### On one extremal problem for positive series

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1677–1683

The approximation properties of the spaces $S^p_{\varphi}$ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of $n$-term approximations of $q$-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set. Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way.

### On Saturation of Linear Summation Methods for Fourier Series in the Spaces $S_{\varphi} ^p$

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 133-138

We consider the problem of the saturation of linear summation methods for Fourier series in the spaces $S_{\varphi} ^p,\; р > 0$. We show that the saturation of a linear method and the saturation order are independent of the parameters $X, ϕ$, and p that define the space $S_{\varphi} ^p(X)$.

### Best $n$-Term Approximations by Λ-Methods in the Spaces $S_ϕ^p$

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1107

We determine the exact values of the upper bounds of $n$-term approximations of $q$-ellipsoids by Λ-methods in the spaces $S_ϕ^p$ in the metrics of the spaces $S_ϕ^p$.