2019
Том 71
№ 11

Romanyuk A. S.

Articles: 41
Article (Ukrainian)

Estimates of some approximating characteristics of the classes of periodic functions of one and many variables

Ukr. Mat. Zh. - 2019. - 71, № 8. - pp. 1102-1115

UDC 517.5
We obtain the exact-order estimates for some approximating characteristics of the classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}$ of periodic functions of one and many variables in the norm of the space $B_{\infty, 1}.$

Anniversaries (Ukrainian)

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

Article (Ukrainian)

Approximating characteristics of the classes of periodic multivariate functions in the space $B_{∞,1}$

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 271-282

We obtain the exact-order estimates of the Kolmogorov widths and entropy numbers for the classes $W^{r}_{p,\alpha}$ and $B^r _{p,\theta}$ in the norm of the $B_{\infty ,1} -space. Article (Ukrainian) Kolmogorov widths and bilinear approximations of the classes of periodic functions of one and many variables Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 224-235 We obtain the exact order estimates for the Kolmogorov widths of the classes$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 Article (Russian) Trigonometric widths of the classes$B_{p,θ}^ r$of functions of many variables in the space$L_q$Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1089-1097 We obtain estimates exact in order for the trigonometric widths of the Besov classes$B_{p,θ}^ r$of periodic functions of many variables in the space$L_q$for$1 ≤ p ≤ 2 < q < p/(p - 1)$. 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 (Russian) Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 80-89 In the spaceL q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB 1,θ r andB p r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross. Article (Russian) Best trigonometric and bilinear approximations for the Besov classes of functions of many variables Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1097–1111 We obtain order estimates for the best trigonometric and bilinear approximations for the classesB p,θ r of functions of many variables. Article (Russian) On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 79–92 Order estimates are obtained for the best approximations of the classesB 1, θ r in the spaceL q with 1<q<∞ and classesB ∞, θ r in a uniform metric. The behavior of Kolmogorov widths of the classesB p, θ r ,1<p≤∞, in the metric of L is studied. Article (Russian) 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 We study the Kolmogorov widths of Besov classes$B^r_{p, \theta}$of periodic functions of many variables with low smoothness in the space$L_q, 1 < q < ∞$. We also investigate the behavior of widths of such classes with critical indices of smoothness. Article (Russian) 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 The order estimates are obtained for the best trigonometric and bilinear approximations of the classes$B^r_{p, \theta}$of functions of many variables with respect to the metric$L_q$when$p$and$q$satisfy certain relations. Article (Russian) 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 The order estimates for the best trigonometric approximations and the Kolmogorov diameters of the classes$B^r_{p, \theta}$of functions of many variables in the space$L_q$are obtained for certain values of the parameters$p, q\$.

Chronicles (Ukrainian)

The School: “Fourier Series: Theory and Applications”

Ukr. Mat. Zh. - 1993. - 45, № 3. - pp. 879

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