2018
Том 70
№ 6

All Issues

Romanyuk A. S.

Articles: 33
Article (Ukrainian)

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

Romanyuk A. S.

↓ Abstract

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)

Romanyuk A. S., Romanyuk V. S., Samoilenko A. M., Savchuk V. V., Serdyuk A. S., Sokolenko I. V.

Full text (.pdf)

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

Article (Russian)

Trigonometric and linear widths for the classes of periodic multivariable functions

Romanyuk A. S.

↓ Abstract

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

Romanyuk A. S.

↓ Abstract

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

Romanyuk A. S., Romanyuk V. S.

↓ Abstract

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

Romanyuk A. S.

↓ Abstract

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)

Babenko V. F., Davydov O. V., Kofanov V. A., Parfinovych N. V., Pas'ko A. N., Romanyuk A. S., Ruban V. I., Samoilenko A. M., Shevchuk I. A., Shumeiko A. A., Timan M. P., Trigub R. M., Vakarchuk S. B., Velikin V. L.

Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract

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

Romanyuk A. S., Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

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)

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

Full text (.pdf)

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)

Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sokolenko I. V.

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

Anniversaries (Ukrainian)

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.

Full text (.pdf)

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

Article (Russian)

Best bilinear approximations of functions from Nikolskii-Besov classes

Romanyuk A. S., Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

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

Romanyuk A. S., Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

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

Romanyuk A. S., Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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)

Romanyuk A. S., Samoilenko A. M.

Full text (.pdf)

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

Obituaries (Ukrainian)

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.

Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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)

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.

Full text (.pdf)

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

Article (Russian)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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 $L_q. I$

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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 $L_q$

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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

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

Full text (.pdf)

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

Chronicles (Ukrainian)

The second school “Fourier series. Theory and Applications”

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

Full text (.pdf)

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

Romanyuk A. S.

↓ Abstract   |   Full text (.pdf)

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 $L_q$

Romanyuk A. S.

Full text (.pdf)

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

Romanyuk A. S.

Full text (.pdf)

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

Romanyuk A. S.

Full text (.pdf)

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

Romanyuk A. S.

Full text (.pdf)

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