2019
Том 71
№ 9

All Issues

Article (Russian)

Algebraic polynomials least deviating from zero in measure on a segment

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 291–300

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an analogous problem with respect to the integral functionals $∫_{–1}^1 φ (∣f (x)∣) dx$ for functions $φ$ that are defined, nonnegative, and nondecreasing on the semiaxis $[0, +∞)$.

Article (English)

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

Ukr. Mat. Zh. - 2010νmber=5. - 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 Hankel determinants of functions given by their expansions in $P$-fractions

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 315–326

We obtain explicit formulas that express the Hankel determinants of functions given by their expansions in continued $P$-fractions in terms of the parameters of the fraction. As a corollary, we obtain a lower bound for the capacity of the set of singular points of these functions, an analog of the van Vleck theorem for $P$-fractions with limit-periodic coefficients, another proof of the Gonchar theorem on the Leighton conjecture, and an upper bound for the radius of the disk of meromorphy of a function given by a $С$-fraction.

Article (English)

On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 327–331

A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.

Article (Russian)

On one result of J. Bourgain

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 332–368

In a linear space of dimension $n$ over the field $\mathbb{F}_2$, we construct a set $A$ of given density such that the Fourier transform of $A$ is large on a large set, and the intersection of $A$ with any subspace of small dimension is small. The results obtained show, in a certain sense, the sharpness of one theorem of J. Bourgain.

Article (English)

Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 369–386

In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters.

Article (Russian)

Quantitative form of the Luzin $C$-property

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 387–395

We prove the following statement, which is a quantitative form of the Luzin theorem on $C$-property: Let $(X, d, μ)$ be a bounded metric space with metric $d$ and regular Borel measure $μ$ that are related to one another by the doubling condition. Then, for any function $f$ measurable on $X$, there exist a positive increasing function $η ∈ Ω\; \left(η(+0) = 0\right.$ and $η(t)t^{−a}$ decreases for a certain $\left. a > 0\right)$, a nonnegative function $g$ measurable on $X$, and $a$ set $E ⊂ X, μE = 0$, for which $$|f(x)−f(y)| ⩽ [g(x)+g(y)]η(d(x,y)),\;x,y ∈ X \setminus E.$$ If $f ∈ L^p(X),\; p >0$, then it is possible to choose $g$ belonging to $L^p (X)$.

Article (English)

Best approximation by ridge functions in $L_p$-spaces

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 396–408

We study the approximation of the classes of functions by the manifold $R_n$ formed by all possible linear combinations of $n$ ridge functions of the form $r(a · x))$. It is proved that, for any $1 ≤ q ≤ p ≤ ∞$, the deviation of the Sobolev class $W^r_p$ from the set $R_n$ of ridge functions in the space $L_q (B^d)$ satisfies the sharp order $n^{-r/(d-1)}$.

Article (Russian)

One-sided approximation of a step by algebraic polynomials in the mean

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 409–422

An asymptotically sharp estimate is obtained for the best one-sided approximation of a step by algebraic polynomials in the space $L_1$.

Article (Russian)

On relative widths of classes of differentiable functions. II

Ukr. Mat. Zh. - 2010νmber=5. - 62, № 3. - pp. 423–431

We obtain an upper bound for the least value of the factor $М$ for which the Kolmogorov widths $d_n (W_C^r, C)$ are equal to the relative widths $K_n (W^C_r, MW^C_j, C)$ of the class of functions $W_C^r$ with respect to the class $MW^C_j$, provided that $j > r$. This estimate is also true in the case where the space $L$ is considered instead of $C$.