2018
Том 70
№ 9

# Romanyuk A. S.

Articles: 33
Article (Ukrainian)

### Kolmogorov widths and bilinear approximations of the classes of periodic functions of one and many variables

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 224-235

We obtain the exact order estimates for the Kolmogorov widths of the classes $W^g_p$ of periodic functions of one variable generated by the integral operators with kernels $g(x, y)$ from the Nikol’skii – Besov classes $B^r_{p,\theta}$. We also study the behavior of bilinear approximations to the classes $W^r_{p,\alpha}$ of periodic multivariate functions with bounded mixed derivative in the spaces $L_{q_1,q_2}$ for some relations between the parameters $r_1, p, q_1, q_2$.

Anniversaries (Ukrainian)

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

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

Article (Russian)

### Trigonometric and linear widths for the classes of periodic multivariable functions

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 670-681

We establish the exact-order estimates for the trigonometric widths of Nikol’skii – Besov $B^r_{\infty ,\theta}$ and Sobolev $W^r_{\infty, \alpha}$ classes of periodic multivariable functions in the space $L_q,\; 1 < q < \infty$. The behavior of the linear widths of Nikol’skii – Besov $B^r_{\infty ,\theta}$ classes in the space $L_q$ is investigated for certain relations between the parameters $p$ and $q$.

Article (Russian)

### Entropy numbers and the widths of the classes $B_{p,θ}^r$ of periodic functions of many variables

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1403-1417

We establish the exact order estimates for the entropy numbers of the Nikol’skii – Besov classes of periodic functions of two variables in the space $L_\infty$ and use the obtained results in estimating the lower bounds for Kolmogorov, linear, and trigonometric widths. We also study the behavior of similar approximating characteristics of the classes $B_{p,θ}^r$ of periodic functions of many variables in the spaces $L_1$ and $B_{1,1}$.

Article (Russian)

### Estimates for the best bilinear approximations of the classes $B^r_{p,\theta}$ and singular numbers of integral operators

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1240-1250

We obtain the exact-order estimates for the best bilinear approximations of the Nikol‘ski–Besov classes $B^r_{p,\theta}$ of periodic functions of several variables. We also find the orders for singular numbers of the integral operators with kernels from the classes $B^r_{p,\theta}$.

Article (Russian)

### Estimation of the Entropy Numbers and Kolmogorov Widths for the Nikol’skii–Besov Classes of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1540-1556

We establish order estimates for the entropy numbers of the Nikol’skii–Besov classes $B_{p,θ}^r$ of periodic functions of many variables in the space $L_q$ with certain relations between the parameters $p$ and $q$. By using the obtained lower estimates of the entropy numbers, we establish the exact-order estimates for the Kolmogorov widths of the same classes of functions in the space $L_1$.

Anniversaries (Ukrainian)

### Motornyi Vitalii Pavlovych (on his 75th birthday)

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999

Article (Russian)

### On the Problem of Linear Widths of the Classes $B_{p,θ}^r$ of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 970–982

We establish order estimates for the linear widths of the classes $B_{p,θ}^r$ of periodic functions of many variables in the space $L_q$ for some relationships between the parameters $p, q$, and $θ$.

Article (Russian)

### Best Bilinear Approximations for the Classes of Functions of Many Variables

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1681–1699

We obtain upper bounds for the values of the best bilinear approximations in the Lebesgue spaces of periodic functions of many variables from the Besov-type classes. In special cases, it is shown that these bounds are order exact.

Anniversaries (Ukrainian)

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

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

Chronicles (Ukrainian)

### 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)

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

Anniversaries (Ukrainian)

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

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

Article (Russian)

### Best bilinear approximations of functions from Nikolskii-Besov classes

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 685-697

We obtain exact-order estimates for the best bilinear approximations of Nikol'skii-Besov classes in the spaces of functions $L_q (\pi_{2d})$.

Article (Russian)

### Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 536–551

We obtain exact order estimates for the best $M$-term trigonometric approximations of the Besov classes $B_{∞,θ}^r$ in the space $L_q$. We also determine the exact orders of the best bilinear approximations of the classes of functions of $2d$ variables generated by functions of d variables from the classes $B_{∞,θ}^r$ with the use of translation of arguments.

Article (Russian)

### Trigonometric and orthoprojection widths of classes of periodic functions of many variables

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1348-1366

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes $B^r_{p,θ}$ and Nikol’skii classes $Hr p$ of periodic functions of many variables in the space $L_q$ for certain relations between the parameters $p$ and $q$.

Article (Russian)

### Approximative characteristics of the isotropic classes of periodic functions of many variables

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 513-523

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov $(B_{p,θ}^r)$ and Nukol’skii $(H_p^r )$ classes of periodic functions of many variables in the metric of $L_q , 1 ≤ p, q ≤ ∞$. We also establish the orders of the best approximations of functions from the same classes in the spaces $L_1$ and $L_{∞}$ by trigonometric polynomials with the corresponding spectrum.

Chronicles (Ukrainian)

### Bogolyubov Readings-2008. International Conference "Differential Equations, Theory of Functions and Applications" (on the occasion of the 70th anniversary of academician AM Samoilenko)

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1722

Obituaries (Ukrainian)

### Alexander Ivanovich Stepanets

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

Article (Russian)

### Best approximations of the classes $B_{p,\,\theta}^{r}$ of periodic functions of many variables in uniform metric

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1395–1406

We obtain estimates exact in order for the best approximations of the classes $B_{\infty,\,\theta}^{r}$ of periodic functions of two variables in the metric of $L_{\infty}$ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes $B_{p,\,\theta}^{r},\quad 1 \leq p < \infty$, of periodic functions of many variables in the metric of $L_{\infty}$ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross.

Anniversaries (Ukrainian)

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

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

Article (Russian)

### Approximation of Classes $B_{p,θ}^r$ by Linear Methods and Best Approximations

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 670-680

We investigate problems related to the approximation by linear methods and the best approximations of the classes $B_{p,{\theta }}^r ,\; 1 ≤ p ≤ ∞$ in the space $L_{∞}$.

Article (Russian)

### Estimates for Approximation Characteristics of the Besov Classes $B_ r^{p,θ}$ of Periodic Functions of Many Variables in the Space $L_q. I$

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1224-1231

We obtain order estimates for the approximation of the classes $B_ r^{p,θ}$ of periodic functions of many variables in the space $L_q$ by using operators of orthogonal projection and linear operators satisfying certain conditions.

Brief Communications (Russian)

### К вопросу об оценках колмогоровских поперечников классов $B_{p,q}^r$ в пространстве $L_q$

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 996-1001

We obtain an order estimate for the Kolmogorov width of the Besov classes $B_{p,{\theta }}^r$ of periodic functions of many variables in the space $L_q$ for $2 < p < q < ∞$, which complements the result obtained earlier by the author.

Article (Russian)

### Linear Widths of the Besov Classes of Periodic Functions of Many Variables. II

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 820-829

We obtain order estimates for linear widths of the Besov classes $B_{p,{\theta }}^r$ of periodic functions of many variables in the space L q for certain values of parameters p and q different from those considered in the first part of the work.

Article (Russian)

### Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 647-661

We obtain order estimates for linear widths of the Besov classes $B_{p,\theta}^r$ of periodic functions of many variables in the space L q for certain values of the parameters p and q.

Chronicles (Ukrainian)

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

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

Chronicles (Ukrainian)

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

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

Article (Russian)

### On estimates of approximation characteristics of the Besov classes of periodic functions of many variables

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1250–1261

We obtain order estimates for some approximate characteristics of the Besov classes B p,ϑ r of periodic functions of many variables.

Article (Ukrainian)

### On Kolmogorov widths of classes $B^r_{p, \theta}$ of periodic functions of many variables with low smoothness in the space $L_q$

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 915–926

Article (Ukrainian)

### The best trigonometric and bilinear approximations for functions of many variables from the classes $B^r_{p, \theta}$. II

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1411–1423

Article (Ukrainian)

### The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 663–675

Article (Ukrainian)

### Approximation of the Besov classes of periodic functions of several variables in a space Lq

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1398–1408