2018
Том 70
№ 5

# Romanyuk V. S.

Articles: 15
Anniversaries (Ukrainian)

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

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

Article (Russian)

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

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

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

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

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

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

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

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

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 $q$.

Anniversaries (Ukrainian)

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

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

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

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

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.