# Serdyuk A. S.

### Vladyslav Kyrylovych Dzyadyk (on his 100th birthday)

Dzyadyk Yu. V., Golub A. P., Kovtunets V. V., Letychevs’kyi O. A., Lukovsky I. O., Makarov V. L., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zadiraka V. K.

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 147-150

### 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 $r (r/n \rightarrow \infty , n \rightarrow \infty )$.

### International conference on the mathematical analysis, differential equations, and their applications (MADEA-8) devoted to the 80th birthday of Academician A. M. Samoilenko

Abdullayev F. G., Samoilenko A. M., Savchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1439-1440

### 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

### Approximation of the classes of generalized Poisson integrals by Fourier sums in metrics of the spaces $L_s$

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 695-704

In metrics of the spaces $L_s,\; 1 \leq s \leq \infty$, we establish asymptotic equalities for the upper bounds of approximations by Fourier sums in the classes of generalized Poisson integrals of periodic functions that belong to the unit ball of space $L_1$.

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II

Bodenchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1011-1018

It is shown that the lower bounds of the Kolmogorov widths $d_{2n}$ in the space $C$ established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),\kern1em h>0,\kern1em \beta \in \mathbb{R},}$$ with functions $φ ⊥ 1$ from the unit ball in the space $L_{∞}$ coincide (for all $n ≥ nh$) with the best uniform approximations of these classes by trigonometric polynomials whose order does not exceed $n − 1$. As a result, we obtain the exact values of widths for the indicated classes of convolutions. Moreover, for all $n ≥ nh$, we determine the exact values of the Kolmogorov widths $d_{2n-1}$ in the space $L_1$ of classes of the convolutions of functions $φ ⊥ 1$ from the unit ball in the space $L_1$ with the kernel $H_{h,β}$.

### Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 916–936

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of $2π$-periodic functions whose $(ψ, β)$-derivatives belong to unit balls in the spaces $L_p,\; 1 ≤ p < ∞$, in the case where the sequence $ψ(k)$ is such that the product $ψ(n)n^{1/p}$ may tend to zero slower than any power function and $∑^{∞}_{k=1} ψ^{p′}(k)k^{p′−2} < ∞$ for $1 < p < ∞,\; 1\p+1\p′ = 1$, or $∑^{∞}_{k=1} ψ(k) < ∞$ for $p = 1$. Similar estimates are also established in the $L_s$-metrics, $1 < s ≤ ∞$, for the classes of summable $(ψ, β)$-differentiable functions such that $‖f_{β}^{ψ} ‖1 ≤ 1$.

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

Bodenchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 719-738

We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.

### Order Estimates for the Best Approximations and Approximations by Fourier Sums in the Classes of Convolutions of Periodic Functions of Low Smoothness in the Uniform Metric

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1658–1675

We obtain the exact-order estimates for the best uniform approximations and uniform approximations by Fourier sums in the classes of convolutions of periodic functions from the unit balls of the spaces $L_p, 1 ≤ p < ∞$, with generating kernel $Ψ_{β}$ for which the absolute values of its Fourier coefficients $ψ(k)$ are such that $∑_{k = 1}^{∞} ψ_p ′(k)k^{p ′ − 2} < ∞,\; \frac 1p + \frac 1{p′} = 1$, and the product $ψ(n)n^{1/p}$ cannot tend to zero faster than power functions.

### Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1244–1256

We establish uniform (with respect to the parameter *p*, 1 ≤ *p* ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes *C* _{ β,p } ^{ ψ } of periodic functions generated by sequences *ψ*(*k*) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function *ψ*. Similar estimations are obtained for the best approximations of the classes *L* _{ β,1} ^{ ψ } in metrics of the spaces *L* _{ s }, 1 ≤ *s* ≤ ∞.

### Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1186–1197

We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes *C* ^{ψ} _{β, p } of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces *L* _{ p }, 1 ≤ *p* < ∞, with generating fixed kernels Ψ_{β} ⊂ *L* _{ p′}, \( \frac{1}{p}+\frac{1}{{p^{\prime}}}=1 \) , whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the *L* _{ p } -metric, 1 < *p* ≤ ∞, for the classes *L* ^{ψ} _{β,1} of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψ_{β} ⊂ *L* _{ p } with functions from the unit ball in the space *L* _{1}. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.

### Major Pylypovych Timan (on his 90th birthday)

Babenko V. F., Motornyi V. P., Peleshenko B. I., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Trigub R. M., Vakarchuk S. B.

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1141-1144

### Lebesgue-type inequalities for the de la Vallée-poussin sums on sets of entire functions

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 642–653

For functions from the sets *C* ^{ψ} _{β} *L* _{ s }, 1 ≤ *s* ≤ ∞,
where ψ(*k*) > 0 and \( {\lim_{{k\to \infty }}}\frac{{\psi \left( {k+1} \right)}}{{\psi (k)}} \) , we obtain asymptotically sharp estimates for the norms of deviations of the de la Vallée-Poussin sums in the uniform metric represented in terms of the best approximations of the (ψ, β) -derivatives of functions of this kind by trigonometric polynomials in the metrics of the spaces *L* _{ s }. It is shown that the obtained estimates are sharp on some important functional subsets.

### Lebesgue-type inequalities for the de la Valee-Poussin sums on sets of analytic functions

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 522-537

For functions from the sets $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s,\; 1 ≤ s ≤ ∞$ generated by sequences $ψ(k) > 0$ satisfying the d’Alembert condition $\lim_{k→∞}\frac{ψ(k + 1)}{ψ(k)} = q,\; q ∈ (0, 1)$, we obtain asymptotically unimprovable estimates for the deviations of de la Vallee Poussin sums in the uniform metric in terms of the best approximations of the $(ψ, β)$-derivatives of functions of this sort by trigonometric polynomials in the metrics of the spaces $L_s$. It is proved that the obtained estimates are unimprovable in some important functional subsets of $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s$.

### 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

### Oleksandr Ivanovych Stepanets’ (on the 70 th anniversary of his birthday)

Gorbachuk M. L., Lukovsky I. O., Makarov V. L., Motornyi V. P., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sharko V. V., Zaderei P. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 579-581

### Approximation by interpolation trigonometric polynomials on classes of periodic analytic functions

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 698-712

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the sets $C^{\psi}_{\beta}L_p$ of $(\psi, \beta)$-differentiable functions generated by sequences $\psi(k)$ that satisfy the d'Alembert conditions. We find asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials on the classes $C^{\psi}_{\beta, p},\;\; 1 \leq p \leq \infty$.

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

Chaichenko S. O., Serdyuk A. S.

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.

### Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1672–1686

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces $L_s , 1 ≤ s ≤ ∞$, we establish asymptotic equalities for upper bounds of approximations by de la Vallée-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallée-Poussin sums in the metrics of the spaces $L_s , 1 ≤ s ≤ ∞$, on the classes of Poisson integrals of functions belonging to the unit ball of the space $L_1$.

### 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.

### Approximation of the classes $C_{β}^{ψ} H_{ω}$ by generalized Zygmund sums

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 524-537

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set $H_{ω}$ in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.

### 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.

### Approximation of Poisson integrals by one linear approximation method in uniform and integral metrics

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 976–982

We obtain asymptotic equalities for the least upper bounds of approximations of classes of Poisson integrals of periodic functions by a linear approximation method of special form in the metrics of the spaces *C* and *L _{p }*.

### Alexander Ivanovich Stepanets

Gorbachuk M. L., Lukovsky I. O., Mitropolskiy Yu. A., Romanyuk A. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1722-1724

### 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.

### Approximation of classes of analytic functions by Fourier sums in the metric of the space $L_p$

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1395–1408

Asymptotic equalities are established for upper bounds of approximants by Fourier partial sums in a metric of spaces $L_p,\quad 1 \leq p \leq \infty$ on classes of the Poisson integrals of periodic functions belonging to the unit ball of the space $L_1$. The results obtained are generalized to the classes of $(\psi, \overline{\beta})$-differentiable functions (in the Stepanets sense) that admit the analytical extension to a fixed strip of the complex plane.

### Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1079 – 1096

We find asymptotic equalities for upper bounds of approximations by Fourier partial sums in a uniform metric on classes of Poisson integrals of periodic functions belonging to unit balls of spaces $L_p,\quad 1 \leq p \leq \infty$. We generalize the results obtained to classes of $(\psi, \overline{\beta})$-differentiable functions (in the Stepanets sense) that admit analytical extension to a fixed strip of the complex plane.

### Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 946–971

We consider classes of $2\pi$-periodic functions that are representable in terms of convolutions with fixed kernels $\Psi_{\overline{\beta}}$ whose Fourier coefficients tend to zero with the exponential rate. We compute exact values of the best approximations of these classes of functions in a uniform and an integral metrics. In some cases, the results obtained enable us to determine exact values of the Kolmogorov, Bernstein, and linear widths for the classes considered in the metrics of spaces $C$ and $L$.

### Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 495–505

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions *C* _{Ψ} ^{β} *C* whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes *C* _{Ψ} ^{β,∞} and *C* _{Ψ} ^{β} *H* _{ω}.

### Approximation of Poisson Integrals by de la Vallée Poussin Sums

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 97-107

On the classes of Poisson integrals of functions belonging to unit balls in the spaces *C* and *L*, we obtain asymptotic equalities for the upper bounds of approximations by de la Vallée Poussin sums in the uniform metric and the integral metric, respectively.

### Oleksandr Ivanovych Stepanets' (on his 60-th birthday)

Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Romanyuk A. S., Romanyuk V. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 579-580

### Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials in the Metric of the Space $L$

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 692-699

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials in the metric of the space *L* on classes of convolutions of periodic functions admitting a regular extension into a fixed strip of the complex plane.

### Direct and Inverse Theorems in the Theory of Approximation of Functions in the Space $S^p$

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 106-124

We continue the investigation of approximation properties of the space $S^p$. We introduce the notion of kth modulus of continuity and establish direct and inverse theorems on approximation in the space $S^p$ in terms of the best approximations and moduli of continuity. These theorems are analogous to the well-known theorems of Jackson and Bernshtein.

### Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1654-1663

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of *L* _{1} on the classes of convolutions.

### Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1689-1701

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials on classes of convolutions of periodic functions admitting a regular extension to a fixed strip of the complex plane.

### Approximation of Periodic Functions of High Smoothness by Interpolation Trigonometric Polynomials in the Metric of $L_1$

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 994-998

We establish an asymptotically exact estimate for the error of approximation of ℝ^{2}-periodic functions of high smoothness by interpolation trigonometric polynomials in the metric of *L* _{1}.

### Lebesgue inequalities for poisson integrals

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 798–808

We obtain estimates for the deviations of the Fourier partial sums on the sets of the Poisson integrals of functions from the space*L* _{ p },*p*≥1, that are expressed in terms of the values of the best approximations of such functions by trigonometric polynomials in the metric of*L* _{ p }. We show that the estimates obtained are unimprovable on some important functional subsets.

### Approximation by fourier sums and best approximations on classes of analytic functions

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 375-395

We establish asymptotic equalities for upper bounds of approximations by Fourier sums and for the best approximations in the metrics of *C* and *L1* on classes of convolutions of periodic functions that can be regularly extended into a fixed strip of the complex plane.

### International conference on the theory of approximation of functions and its applications dedicated to the memory of V. K. Dzyadyk

Romanyuk A. S., Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1296–1297

### Widths and best approximations for classes of convolutions of periodic functions

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 674–687

We establish exact lower bounds for the Kolmogorov widths in the metrics of*C* and*L* for classes of functions with high smoothness; the elements of these classes are sourcewise-representable as convolutions with generating kernels that can increase oscillations. We calculate the exact values of the best approximations of such classes by trigonometric polynomials.

### On the existence and uniqueness of a solution of the problem of uniform *SK*-spline-interpolation

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 486–492

We establish necessary and sufficient conditions for the existence and uniqueness of generalized interpolating*SK*-splines with a uniform distribution of interpolation points.

### Estimates of the Kolmogorov widths for classes of infinitely differentiable periodic functions

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1700–1706

Lower estimates of the Kolmogorov widths are obtained for certain classes of infinitely differentiable periodic functions in the metrics of *C* and *L.* For many important cases, these estimates coincide with the values of the best approximations of convolution classes by trigonometric polynomials calculated by Nagy, and, hence, they are exact.

### The second school “Fourier series. Theory and Applications”

Romanyuk A. S., Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1584