# Volume 70, № 9, 2018

### Jackson – Stechkin-type inequalities for the approximation of elements of Hilbert spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1155-1165

We introduce new characteristics for elements of Hilbert spaces, namely, generalized moduli of continuity \$\omega_{ \varphi} (x, L_p, V ([0, \delta]))$ and obtain new exact Jackson – Stechkin-type inequalities with these moduli of continuity for the approximation of elements of Hilbert spaces. These results include numerous well-known inequalities for the approximation of periodic functions by trigonometric polynomials, approximation of nonperiodic functions by entire functions of exponential type, similar results for almost periodic functions, etc. Some of these results are new even in these classical cases.

### Generalized characteristics of smoothness and some extremе problems of the approximation theory of functions in the space $L_2 (R)$. I

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1166-1191

We consider the generalized characteristics of smoothness of the functions $\omega^w(f, t)$ and $\Lambda^w(f, t), t > 0,$ in the space $L_2(R)$ and, on the classes $L^{\alpha}_2 (R)$ defined with the help of fractional-order derivatives $\alpha \in (0,\infty)$, obtain the exact Jackson-type inequalities for $\omega^w(f)$.

### Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1192-1210

We consider the approximation of smooth functions by two weighted $N$-point Pad´e approximants. We present numerical examples and the inequalities between the Stietjes function and its $N$-point Padé approximant. Квартиры посуточно - онлайн-бронирование

### Some results on the global solvability for structurally damped models with a special nonlinearity

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1211-1231

The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem $$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$. The parameter $\delta \in (0, \sigma]$ describes the structural damping in the model varying from the exterior damping $\delta = 0$ up to the visco-elastic type damping $\delta = \sigma$. We will obtain the admissible sets of the parameter p for the global solvability of this semilinear Cauchy problem with arbitrary small initial data u0, u1 in the hyperbolic-like case $\delta \in \Bigl(\cfrac{\sigma}{2} , \sigma \Bigr)$, and in the exceptional case $\delta = 0$.

### First nontrivial group of homologies of simplicial schemes of unimodular frames over the Dedekind ring

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1231-1248

We prove a theorem on generation of the first nontrivial group of homologies of a simplicial scheme of unimodular frames over the Dedekind ring by standard cycles.

### Estimates for the entropy numbers of the classes $B_{p,θ}^{Ω}$ of periodic multivariate functions in the uniform metric

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1249-1263

We establish order estimates for the entropy numbers of the classes $B_{p,θ}^{Ω}$ of periodic multivariate functions in the uniform metric. For the proper choice of the functions $\Omega$, these classes coincide with the Nikol’skii – Besov classes $B_{p,θ}^{r}$.

### On the equicontinuity of one family of inverse mappings in terms of prime ends

Salimov R. R., Sevost'yanov E. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1264-1273

For a class of mappings satisfying upper modular estimates with respect to families of curves, we study the behavior of the corresponding inverse mappings. In the terms of prime ends, we prove that the families of these homeomorphisms are equicontinuous (normal) in the closure of a given domain.

### Third boundary-value problem for a third-order differential equation with multiple characteristics

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1274-1281

We prove the unique solvability of the third boundary-value problem for a third-order differential equation with multiple characteristics containing the second time derivative in a rectangular domain.

### Inequalities for inner radii of symmetric disjoint domains

Bakhtin A. K., Denega I. V., Vyhovs'ka L.V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1282-1288

We study the following problem: Let $a_0 = 0, | a_1| = ... = | a_n| = 1,\; a_k \in B_k {\subset C}$, where $B_0, ... ,B_n$ are disjoint domains, and $B_1, ... ,B_n$ are symmetric about the unit circle. It is necessary to find the exact upper bound for $r^{\gamma} (B_0, 0) \prod^n_{k=1} r(B_k, a_k)$, where $r(B_k, a_k)$ is the inner radius of Bk with respect to $a_k$. For $\gamma = 1$ and $n \geq 2$, the problem was solved by L. V. Kovalev. We solve this problem for $\gamma \in (0, \gamma_n], \gamma_n = 0,38 n^2$, and $n \geq 2$ under the additional assumption imposed on the angles between the neighboring line segments $[0, a_k]$.

### Coefficient estimates for two subclasses of analytic and bi-univalent functions

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1289-1296

We introduce two new subclasses of the class $\sigma$ of analytic and bi-univalent functions in the open unit disk $U$. Furthermore, we obtain the estimates for the first two Taylor – Maclaurin coefficients $|a_2|$ and $|a_3|$ for the functions from these new subclasses.