2018
Том 70
№ 4

All Issues

Parfinovych N. V.

Articles: 9
Article (Russian)

Exact values of the best (α, β) -approximations of classes of convolutions with kernels that do not increase the number of sign changes

Parfinovych N. V.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1073-1083

We obtain the exact values of the best $(\alpha , \beta )$-approximations of the classes $K \ast F$ of periodic functions $K \ast f$ such that $f$ belongs to a given rearrangement-invariant set $F$ and $K$ is $2\pi$ -periodic kernel that do not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points $2k\pi /n$ and $2k\pi /n + h, n \in N, k \in Z, h \in (0, 2\pi /n)$. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Article (Russian)

Estimation of the uniform norm of one-dimensional Riesz potential of a partial derivative of a function with bounded Laplacian

Babenko V. F., Parfinovych N. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 7. - pp. 867-878

We obtain new exact Landau-type estimates for the uniform norms of one-dimension Riesz potentials of the partial derivatives of a multivariable function in terms of the norm of the function itself and the norm of its Laplacian.

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 (English)

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Babenko V. F., Parfinovych N. V., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 301–314

Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented.

Article (Russian)

On the order of relative approximation of classes of differentiable periodic functions by splines

Babenko V. F., Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 147–157

In the case where $n → ∞$, we obtain order equalities for the best $L_q$ -approximations of the classes $W_p^r ,\; 1 ≤ q ≤ p ≤ 2$, of differentiable periodical functions by splines from these classes.

Article (Russian)

Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

Babenko V. F., Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1443-1454

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.

Brief Communications (Russian)

Inequalities of the Bernstein type for splines of defect 2

Babenko V. F., Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 995-999

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.

Article (Ukrainian)

Exact order of relative widths of classes $W^r_1$ in the space $L_1$

Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1409–1417

As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions.

Article (Russian)

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 489-500

We obtain the exact asymptotics (as n → ∞) of the best L 1-approximations of classes \(W_1^r\) of periodic functions by splines sS 2n, r − 1 and sS 2n, r + k − 1 (S 2n, r is the set of 2π-periodic polynomial splines of order r and defect 1 with nodes at the points kπ/n, k ∈ Z) under certain restrictions on their derivatives.