Parfinovych N. V.
Exact values of the best (α, β) -approximations of classes of convolutions with kernels that do not increase the number of sign changes
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.
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.
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.
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.
Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999
Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions
Babenko V. F., Parfinovych N. V., Pichugov S. A.
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.
On the order of relative approximation of classes of differentiable periodic functions by splines
Babenko V. F., Parfinovych N. V.
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.
Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities
Babenko V. F., Parfinovych N. V.
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.
Inequalities of the Bernstein type for splines of defect 2
Babenko V. F., Parfinovych N. V.
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.
Exact order of relative widths of classes $W^r_1$ in the space $L_1$
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.
On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines
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 s ∈ S 2n, r − 1 and s ∈ S 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.
On the best $L_1$-approximations of functional classes by splines under restrictions imposed on their derivatives
Babenko V. F., Parfinovych N. V.
Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 435-444
We find the exact asymptotics ($n → ∞$) of the best $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s ∈ S_{2n, r∼-1}$ ($S_{2n, r∼-1}$ is a set of $2π$-periodic polynomial splines of order $r−1$, defect one, and with nodes at the points $kπ/n,\; k ∈ ℤ$) such that $V_0^{2π} s^{( r-1)} ≤ 1+ɛ_n$, where $\{ɛ_n\}_{n=1}^{ ∞}$ is a decreasing sequence of positive numbers such that $ɛ_n n^2 → ∞$ and $ɛ_n → 0$ as $n → ∞$.