# Volume 71, № 2, 2019

### Vladyslav Kyrylovych Dzyadyk (on his 100th birthday)

Dzyadyk Yu. V., Holub A. P., Kovtunets V. V., Letychevs’kyi O. A., Lukovsky I. O., Makarov V. L., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zadiraka V. K.

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 147-150

### Application of Dzyadyk’s polynomial kernels in the constructive function theory

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 151-157

This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzjadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.

### One inequality of the Landau – Kolmogorov type for periodic functions of two variable

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 158-167

We obtain a new sharp inequality of the Landau – Kolmogorov type for a periodic function of two variables that estimates the convolution of the best uniform approximations of its partial primitives by the sums of univariate functions with the help of its $L_{\infty}$ -norm and uniform norms of its mixed primitives. Some applications of the obtained inequality are presented.

### Piecewise-polynomial approximations for the solutions of impulsive differential equations

Bilenko V. I., Bozhonok K. V., Dzyadyk S. Yu.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 168-179

On the basis of V. Dzyadyk’s approximation method, we consider the problems of construction and theoretical substantiation of high-precision numerical-analytic algorithms for the piecewise polynomial approximation of the solutions of problems with pulsed action.

### On the estimates of widths of the classes of functions defined by the generalized moduli of continuity and majorants in the weighted space $L_{2x} (0,1)$

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 179-189

The upper and lower estimates for the Kolmogorov, linear, Bernstein, Gelfand, projective, and Fourier widths are obtained in the space $L_{2,x}(0, 1)$ for the classes of functions $W^r_2 (\Omega^{(\nu )}_{m,x}; \Psi )$, where $r \in Z+, m \in N, \nu \geq 0,$ and $\\Omega^{(\nu )}_{m,x}$ and $\Psi$ are the mth order generalized modulus of continuity and the majorant, respectively. The upper and lower estimates for the suprema of Fourier – Bessel coefficients were also found on these classes. We also present the conditions for majorants under which it is possible to find the exact values of indicated widths and the suprema of Fourier – Bessel coefficients.

### Resonant equations with classical orthogonal polynomials. I

Gavrilyuk I. P., Makarov V. L.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 190-209

In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an algorithm of finding their particular and general solutions in the explicit form. The algorithm is especially suitable for the computer algebra tools, such as Maple. The resonant equations form an essential part of various applications e.g. of the efficient functional-discrete method aimed at the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as for the square operator equations $A^2u = f$; e.g., for the biharmonic equation.

### Quasiunconditional basis property of the Faber – Schauder system

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 210-219

We prove that, for any $0 < \delta < 1$, there exists a measurable set $E_{\delta} \subset [0, 1], \mathrm{m}\mathrm{e}\mathrm{s} (E_{\delta }) > 1 \delta $, such that for any function $f \in C[0, 1]$, one can find a function $\widetilde f \in C[0, 1]$ that coincides with f on E\delta , and the Fourier – Faber – Schauder series for the function $\widetilde f$ unconditionally converges in $C[0, 1]$. Moreover, the moduli of the nonzero Fourier – Faber – Schauder coefficients of the function $\widetilde f$ coincide with the elements of a given sequence $\{ b_n\}$ satisfying the condition $$b_n \downarrow 0,\; \sum^{\infty }_{n=1} frac{b_n}{n} = +\infty .$$

### On Cèsaro and Copson norms of nonnegative sequences

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 220-229

The C`esaro and Copson norms of a nonnegative sequence are lp-norms of its arithmetic means and the corresponding conjugate means. It is well known that, for $1 < p < \infty$, these norms are equivalent. In 1996, G. Bennett posed the problem of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four constants. Two of them were found by G. Bennett. We find one of the two unknown constants and also prove one optimal weighted-type estimate regarding the remaining constant.

### On one estimate of divided differences and its applications

Kopotun K. A., Leviatan D., Shevchuk I. A.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 230-245

We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their generalization to the Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f \in C(r)(I)$ and a set $Z = \{ z_j\}^{\mu}_{j=0}$ such that $z_{j+1} - z_j \geq \lambda | I|$ for all $0 \leq j \leq \mu 1$, where $I := [z_0, z_{\mu} ], | I|$ is the length of $I$, and $\lambda$ is a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality $\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$, $$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1} \int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$ where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $.

### Estimation of the rate of decrease (vanishing) of a function in terms of relative oscillations

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 246-260

The relative mean integral oscillations of a nondecreasing equimeasurable rearrangement are estimated from above via the same oscillations of the original function. On the basis of this estimate, we establish a lower order-exact estimate for the rate of decrease (vanishing) of the rearrangement.

### On the joint approximation of a function and its derivatives in the mean

Motornaya O. V., Motornyi V. P.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 261-270

We consider some properties of functions integrable on a segment. Some estimates for the approximations of function and its derivatives are obtained.

### Approximating characteristics of the classes of periodic multivariate functions in the space $B_{∞,1}$

Romanyuk A. S., Romanyuk V. S.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 271-282

We obtain the exact-order estimates of the Kolmogorov widths and entropy numbers for the classes $W^{r}_{p,\alpha}$ and $B^r _{p,\theta}$ in the norm of the $B_{\infty ,1} -space.

### Approximation by interpolation trigonometric polynomials in metrics of the spaces $L_p$ on the classes of periodic entire functions

Serdyuk A. S., Sokolenko I. V.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 283-292

We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with equidistant distribution of interpolation nodes $x_{(n 1)}^k = \frac{2k\pi}{2n 1}, k \in Z,$, in metrics of the spaces $L_p$ on the classes of $2\pi$ -periodic functions that can be represented in the form of convolutions of functions $\varphi , \varphi \bot 1$, from the unit ball of the space $L_1$, with fixed generating kernels in the case where the modules of their Fourier coefficients $\psi (k)$ satisfy the condition $\mathrm{lim}_{k\rightarrow \infty} \psi (k + 1)/\psi (k) = 0.$. Similar estimates are also obtained on the classes of $r$-differentiable functions $W^r_1$ for the rapidly increasing exponents of smoothness $r (r/n \rightarrow \infty , n \rightarrow \infty )$.

### On the approximation of functions by polynomials and entire functions of exponential type

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 2. - pp. 293-300

We present a brief survey of works in the approximation theory of functions known to the author and connected with V. K. Dzyadyk’s scientific publications.