# Stepanets O. I.

### 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.

### 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.

### Multiple Fourier sums and ψ-strong means of their deviations on the classes of ψ-differentiable functions of many variables

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1075–1093

We present results concerning the approximation of ψ-differentiable functions of many variables by rectangular Fourier sums in uniform and integral metrics and establish estimates for φ-strong means of their deviations in terms of the best approximations.

### 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.

### 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

### On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

### Andrei Reuter (1937-2006)

Bondarenko V. M., Drozd Yu. A., Kirichenko V. V., Mitropolskiy Yu. A., Samoilenko A. M., Samoilenko Yu. S., Sharko V. V., Stepanets O. I.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1584-1585

### Problems of approximation theory in linear spaces

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 47–92

We present a survey of results related to the approximation characteristics of the spaces $S^{\rho}_{\varphi}$ and their generalizations. The proposed approach enables one to obtain solutions of problems of classical approximation theory in abstract linear spaces in explicit form. The results obtained yield statements that are new even in the case of approximations in the functional Hilbert spaces $L_2$.

### 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.

### Best $n$-Term Approximations with Restrictions

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 533–553

We determine exact values of the best $n$-term approximations with restrictions on polynomials used for the approximation of $\lambda, q$-ellipsoids in the spaces $S^{p,\, \mu}_{\varphi}$.

### The international conference „International workshop on analysis and its applications"

Samoilenko A. M., Shevchuk I. A., Stepanets O. I.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1722

### Best Approximations of $q$-Ellipsoids in Spaces $S_{ϕ}^{p,μ}$

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1378-1383

We find exact values of the best approximations and basic widths of $q$-ellipsoids in the spaces $S_{ϕ}^{p,μ}$ for $q > p > 0$.

### 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.

### Extremal Problems of Approximation Theory in Linear Spaces

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1378-1409

We propose an approach that enables one to pose and completely solve main extremal problems in approximation theory in abstract linear spaces. This approach coincides with the traditional one in the case of approximation of sets of functions defined and square integrable with respect to a given σ-additive measure on manifolds in *R* ^{m}, *m* ≥ 1.

### 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 $S_ϕ^p$.

### Best “Continuous” $n$-Term Approximations in the Spaces $S_\phi ^p$

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 663-670

We find exact values of upper bounds for the best approximations of $q$-ellipsoids by polynomials of degree $n$ in the spaces $S_\phi ^p$ in the case where the approximating polynomials are constructed on the basis of $n$-dimensional subsystems chosen successively from a given orthonormal system ϕ.

### Spaces $S^p$ with Nonsymmetric Metric

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 264-277

We determine exact values of the best approximations and Kolmogorov widths of $q$-ellipsoids in spaces $S_\phi ^{p,{\mu}}$ defined by anisotropic metric.

### Approximation Properties of the de la Vallée-Poussin Method

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1100-1125

We present a survey of results concerning the approximation of classes of periodic functions by the de la Vallée-Poussin sums obtained by various authors in the 20th century.

### Approximation of Convolution Classes by Fourier Sums. New Results

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 581-602

We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.

### Approximation of Cauchy-Type Integrals

Savchuk V. V., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 706-740

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of ψ-integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded *r*th derivatives (here, *r* is a natural number).

On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of ψ-integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.

### 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 Characteristics of the Spaces $S_p^{ϕ}$ in Different Metrics

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1121-1146

We continue the investigation of the approximation characteristics of the spaces $S_p^{ϕ}$ introduced earlier. In particular, we establish direct and inverse theorems on the approximation of elements of these spaces. We also determine the exact values of upper bounds of $m$-term approximations of $q$-ellipsoids in the spaces $S_p^{ϕ}$ in the metrics of the spaces $S_p^{ϕ}$.

### 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.

### 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.

### Rate of convergence of a group of deviations on sets of $\bar{\psi}$−integrals

Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1673-1693

We study functionals that characterize the strong summation of Fourier series on sets of $\bar{\psi}$−integrals in the uniform and integral metrics. As a result, we obtain estimates exact in order for the best approximations of functions from these sets by trigonometric polynomials.

### Approximation of locally integrable functions on the real line

Stepanets O. I., Wang Kunyang, Zhang Xirong

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1549–1561

We introduce the notion of generalized \(\bar \psi \) -derivatives for functions locally integrable on the real axis and investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type.

### 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

### Several statements for convex functions

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 688–702

For the setM of convex-downward functions Ψ (•) vanishing at infinity, we present its decomposition into subsets with respect to the behavior of special characteristics η (Ψ;•) and μ(Ψ;•) of these functions. We study geometric and analytic properties of the elements of the subsets obtained, which are necessary for the investigation of problems of the theory of approximation for classes of convolutions.

### Approximate properties of the Zygmund method

Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 493–518

We give a review of results on approximate properties of Zygmund sums and their generalizations.

### Approximation of $\bar {\psi} - \text{Integrals}$ of periodic functions by Fourier sums (small smoothness). IIof periodic functions by Fourier sums (small smoothness). II

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 388-400

We investigate the rate of convergence of Fourier series on the classes $L^{\bar {\psi}} - \text{N}$ in the uniform and integral metrics. The results obtained are extended to the case where the classes $L^{\bar {\psi}} - \text{N}$ are the classes of convolutions of functions from $\text{N}$ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets $L^{\bar {\psi}} - \text{N}$, which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.

### Approximation of $\bar {\psi} - integrals$−integrals of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 274-291

We investigate the rate of convergence of Fourier series on the classes $L^{\bar {\psi} } \text{N}$ in the uniform and integral metrics. The results obtained are extended to the case where the classes $L^{\bar {\psi} } \text{N}$ are the classes of convolutions of functions from $\text{N}$ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets $L^{\bar {\psi} } \text{N}$ which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.

### 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

### Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1069-1113

We introduce the notion of $\overline{\psi}$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\overline{\psi}}$. We obtain integral representations of deviations of the trigonometric polynomials $U_{n(f;x;Λ)}$ generated by a given Λ-method for summing the Fourier series of functions $f ε L^{\overline{\psi}}$. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\overline{\psi}}$ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\overline{\psi}}$, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.

### Strong summability of orthogonal expansions of summable functions. II

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 393-405

We study the problem of strong summability of Fourier series in orthonormal systems of polynomialtype functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of the systems under consideration.

### Strong summability of orthogonal expansions of summable functions. I

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 260-277

We study the problem of strong summability of Fourier series in orthonormal systems of polynomial-type functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of systems under consideration.

### Approximations in spaces of locally integrable functions

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 597–625

### Approximation of cauchy-type integrals in Jordan domains

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 809–833

### Multiple Fourier sums on sets of (ψ, β)-differentiable functions

Pachulia N. L., Stepanets O. I.

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 545-555

### Approximation by entire functions in the mean on the real line

Stepanets N. I., Stepanets O. I.

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 121-125

### Approximation of weakly differentiable periodic functions

Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 406-412

### Classes of functions defined on the real axis and their approximations by entire functions. II

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 210-222

### Classes of functions defined on the real line and their approximation by entire functions. I

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 102-112

### Inverse theorems for the approximation of (ψ, β)-Differentiable functions

Stepanets O. I., Zhukina E. I.

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1106–1112

### Deviations of Fourier sums on classes of entire functions

Ukr. Mat. Zh. - 1989. - 41, № 6. - pp. 783–789

### Fourier series: New results and unsolved problems

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 547-562

### Approximation by fourier operators of functions defined on the real line

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 198-209

### Behavior of the group of deviations on sets of (ψ, β)-differentiable functions

Pachulia N. L., Stepanets O. I.

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 101-105

### Mean-square rate of convergence of orthogonal series

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 606–611

### Convergence rate of fourier series and best approximations in the space *L*^{p }

^{p }

Kushpel A. K., Stepanets O. I.

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 483–492

### Approximation by Fourier sums of functions with slowly decreasing Fourier coefficients

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 755–762

### Modules of half-decay of monotonic functions and the rate of convergence of Fourier series

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 618–624

### Approximation of periodic functions by Fourier sums in the mean

Novikova A. K., Stepanets O. I.

Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 204–210

### An order relation for (ψ β)-derivatives

Ukr. Mat. Zh. - 1985. - 37, № 5. - pp. 645–648

### Deviation of Fourier sums on the classes of infinitely differentiable functions

Ukr. Mat. Zh. - 1984. - 36, № 6. - pp. 750 – 758

### Behavior of sequences of partial fourier sums of continuous functions near the points of their divergence

Ukr. Mat. Zh. - 1983. - 35, № 5. - pp. 652—653

### Approximation by triangular fourier sums on classes of continuous periodic functions of two variables

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 1983. - 35, № 2. - pp. 249—254

### Simultaneous approximation of periodic functions and their derivatives by fourier sums

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 356–367

### Asymptotically exact estimates of the errors of partial fourier sums on classes of continuous periodic functions of several variables

Ukr. Mat. Zh. - 1981. - 33, № 1. - pp. 119–123

### Work of A. V. Skorokhod on the theory of stochastic processes

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 528–537

### Solution of an extremal problem for classes of discontinuous functions of two variables in permutations

Ukr. Mat. Zh. - 1979. - 31, № 1. - pp. 95–101

### Approximation of periodic functions of two variables by Vallée-Poussin sums

Stepanets O. I., Zaderei P. V.

Ukr. Mat. Zh. - 1978. - 30, № 1. - pp. 33–44

### Approximation of continuous periodic functions of two variables with the aid of a linear method of interpolational type

Ukr. Mat. Zh. - 1975. - 27, № 1. - pp. 42–61

### Method of approximating continuous periodic functions

Ukr. Mat. Zh. - 1974. - 26, № 6. - pp. 762–774

### Approximation of continuous periodic functions by Rogosinski polynomials

Ukr. Mat. Zh. - 1974. - 26, № 4. - pp. 496–509

### Approximation of certain classes of periodic functions of two variables by linear methods of summation of their Fourier series

Ukr. Mat. Zh. - 1974. - 26, № 2. - pp. 205–215

### The approximation of certain classes of differentiable periodic functions of two variables by Fourier sums

Ukr. Mat. Zh. - 1973. - 25, № 5. - pp. 599—609

### Exact upper bounds of the deviations of Bernstein sums from functions of Halder classes

Gavrilyuk V. T., Stepanets O. I.

Ukr. Mat. Zh. - 1973. - 25, № 2. - pp. 147—157

### Approximation of differentiable functions by Rogosinski polynomials

Gavrilyuk V. T., Stepanets O. I.

Ukr. Mat. Zh. - 1973. - 25, № 1. - pp. 3-13

### Fourier series approximation of functions which satisfy Lipschitz conditions

Ukr. Mat. Zh. - 1972. - 24, № 6. - pp. 781—799

### On a problem of A. N. Kolmogorov for functions of two variables

Ukr. Mat. Zh. - 1972. - 24, № 5. - pp. 653–665

### Asymptotic equations for the supremums of approximations of functions of hölder's classes by rogosinski polynomials

Dzyadyk V. K., Stepanets O. I.

Ukr. Mat. Zh. - 1972. - 24, № 4. - pp. 476–487

### Exact upper bound for approximations on classes of differential periodic functions using Rogosinski polynomials

Dzyadyk V. K., Gavrilyuk V. T., Stepanets O. I.

Ukr. Mat. Zh. - 1970. - 22, № 4. - pp. 481–493

### Approximating quasismooth functions by ordinary polynomials

Polyakov R. V., Stepanets O. I.

Ukr. Mat. Zh. - 1968. - 20, № 4. - pp. 534–540

### Approximation of continuous functions by regular polynomials

Polyakov R. V., Stepanets O. I.

Ukr. Mat. Zh. - 1968. - 20, № 2. - pp. 192–202