2019
Том 71
№ 7

# Konovalov V. N.

Articles: 10
Article (Russian)

### Kolmogorov and linear widths of classes of s-monotone integrable functions

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1633–1652

Let $s \in \mathbb{N}$ and let $\Delta^s_+$ be the set of functions $x \mapsto \mathbb{R}$ on a finite interval $I$ such that the divided differences $[x; t_0, ... , t_s ]$ of order $s$ of these functions are nonnegative for all collections of $s + 1$ distinct points $t_0,..., t_s \in I$. For the classes $\Delta^s_+ B_p := \Delta^s_+ \bigcap B_p$ , where $B_p$ is the unit ball in $L_p$, we obtain orders of the Kolmogorov and linear widths in the spaces $L_q$ for $1 \leq q < p \leq \infty$.

Article (Russian)

### Shape-preserving kolmogorov widths of classes of s-monotone integrable functions

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 901–926

Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences $[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, where $B_p$ is the unit ball of the space $L_p$, and let $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. For every $s \geq 3$ and $1 \leq q \leq p \leq \infty$, exact orders of the shape-preserving Kolmogorov widths $$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$ are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n \leq n$ and $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$.

Article (Russian)

### Approximation of Sobolev Classes by Their Sections of Finite Dimension

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 647-655

For Sobolev classes of periodic functions of one variable with restrictions on higher derivatives in L 2, we determine the exact orders of relative widths characterizing the best approximation of a fixed set by its sections of given dimension in the spaces L q.

Brief Communications (Russian)

### Orders of Trigonometric and Kolmogorov Widths May Differ in Power Scale

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1575 -1579

We present a class of functions for which trigonometric widths decrease to zero slower than the Kolmogorov widths in power scale.

Article (Ukrainian)

### Continuation of functions of several variables with preservation of differential-difference properties

Ukr. Mat. Zh. - 1984. - 36, № 3. - pp. 304 - 308

Article (Ukrainian)

### Approximation of functions of several variables by polynomials with preservation of the differential-difference properties

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 154 - 160

Article (Ukrainian)

### An approximation theorem of the Jackson type for functions of several variables

Ukr. Mat. Zh. - 1981. - 33, № 6. - pp. 757-764

Article (Ukrainian)

### On the best approximations of functions of several variables by multipoint Taylor formulas

Ukr. Mat. Zh. - 1980. - 32, № 1. - pp. 104 - 110

Article (Ukrainian)

### Problem of the diameters of classes of analytic functions

Ukr. Mat. Zh. - 1978. - 30, № 5. - pp. 668–670

Article (Ukrainian)

### Method of expanding unity in regions with piecewise smooth boundaries as sums of algebraic polynomials of two variables having certain properties of a kernel

Ukr. Mat. Zh. - 1973. - 25, № 2. - pp. 179—192