2018
Том 70
№ 11

All Issues

Romanyuk V. S.

Articles: 15
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)

Kolmogorov widths and entropy numbers in the Orlich spaces with the Luxembourg norm

Romanyuk V. S.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 682-694

We obtain the exact-order estimates of the Kolmogorov widths and entropy numbers of unit balls from the binary Besov spaces \$\mathrm{d}\mathrm{y}\mathrm{a}\mathrm{d} B^{0,\gamma}_{ p,\theta}$ compactly embedded in the exponential Orlich $\mathrm{e}\mathrm{x}\mathrm{p} L^{\nu}$ spaces equipped with the Luxembourg norm.

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)

Multiple Haar basis and $m$-term approximations of functions from the Besov classes. II

Romanyuk V. S.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 816-825

We establish the exact-order estimates for the best $m$-term approximations in the multiple Haar basis $\mathrm{H}^d$ of functions from the Besov classes in the Lebesgue spaces $L_q(I^d)$. We also present a practical algorithm of the construction of the extreme nonlinear m-term aggregates (in a sense of the exact-order estimates for approximations).

Article (Russian)

Multiple Haar basis and m-term appriximations for functions from the Besov classes. I

Romanyuk V. S.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 551-562

We describe the isotropic Besov spaces of functions of several variables in the terms of conditions imposed on the Fourier – Haar coefficients.

Article (Ukrainian)

Multiple Haar Basis and its Properties

Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1253–1264

In the Lebesgue spaces $L_p ([0, 1]^d ), 1 ≤ p ≤ ∞$, for $d ≥ 2$, we define a multiple basis system of functions $H^d  = (h_n )_{n = 1}^{∞}$. This system has the main properties of the well-known one-dimensional Haar basis $H$. In particular, it is shown that the system $H^d$ is a Schauder basis in the spaces $L_p ([0, 1]^d ),\; 1 ≤ p < ∞$.

Article (Russian)

Constructive Characteristic of ho¨ Lder Classes and M-Term Approximations in the Multiple Haar Basis

Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 349–360

In terms of the best polynomial approximations in the multiple Haar basis, we obtain a constructive characteristic of the Hölder classes H p α of functions defined on the unit cube \( \mathbb{I} \) d of the space ℝ d under the restriction \( 0<\alpha <\frac{1}{p}\le 1 \) . We also solve the problem of order estimates of the best m-term approximations in the Haar basis of classes H p α in the Lebesgue spaces L q ( \( \mathbb{I} \) d ).

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.

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 $q$.

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 (Ukrainian)

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II

Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 346-355

In normed spaces of functions analytic in the Jordan domain Ω⊂ℂ, we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in Ω by Cauchy-type integrals along Γ = ∂Ω with densities f(·) such that \(f \circ \Psi \in L_{\beta ,p}^\Psi (T)\) . Here, Ψ is a conformal mapping of \(C\backslash \overline \Omega \) onto {w: |w| > 1}, and \(L_{\beta ,p}^\Psi (T)\) is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.

Article (Russian)

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 229-237

In the Banach space of functions analytic in a Jordan domain \(\Omega \subset \mathbb{C}\) , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in Ω by Cauchy-type integrals along the rectifiable curve Γ = ∂Ω and can be analytically continued to Ω′ ⊃ Ω or to \(\mathbb{C}\) .

Article (Russian)

Weighted approximation in mean of classes of analytic functions by algebraic polynomials and finite-dimensional subspaces

Romanyuk V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 645–662

We establish estimates for classic approximation quantities for sets from functional spaces (classes of functions analytic in Jordan domains), namely, for the best polynomial approximations and Kolmogorov widths.