# Volume 64, № 9, 2012

### On convolution of functions in angular domains

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1155-1164

We obtain analogs of the Parseval theorem, convolution theorem, and some other properties of the convolution of functions from the Hardy – Smirnov spaces in an arbitrary convex unbounded polygon.

### Asymptotic representations of solutions of essentially nonlinear systems of ordinary differential equations with regularly and rapidly varying nonlinearities

Evtukhov V. M., Shlepakov O. R.

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1185

We obtain asymptotic representations for one class of solutions of systems of ordinary differential equations more general than systems of the Emden – Fowler type.

### Homotopic types of right stabilizers and orbits of smooth functions on surfaces

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1204

Let $\mathcal{M}$ be a smooth connected compact surface, $P$ be either the real line $\mathbb{R}$ or a circle $S^1$. For a subset $X ⊂ M$, let $\mathcal{D}(M, X)$ denote the group of diffeomorphisms of $M$ fixed on $X$. In this note, we consider a special class F of smooth maps $f : M → P$ with isolated singularities that contains all Morse maps. For each map $f ∈ \mathcal{F}$, we consider certain submanifolds $X ⊂ M$ that are “adopted” with $f$ in a natural sense, and study the right action of the group $\mathcal{D}(M, X)$ on $C^{∞}(M, P)$. The main result describes the homotopy types of the connected components of the stabilizers $S(f)$ and orbits $\mathcal{O}(f)$ for all maps $f ∈ \mathcal{F}$. It extends previous results of the author on this topic.

### Approximation of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables by Fourier sums in the space $L_p$ with $p = 1, \infty$

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1204-1213

We obtain an exact-order estimate for the deviation of Fourier sums of periodic functions of many variables from the classes $B^{\Omega}_{p, \theta}$ in the space $L_p$ for $p = 1, \infty$.

### Smoothness of functions in the metric spaces *L*_{ψ}

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1214-1232

Let $L_0(T)$ be thе set of real-valued periodic measurable functions, let $\psi : R^+ \rightarrow R^+$ be a modulus of continuity $(\psi \neq 0)$ , and let $$L_{\psi} \equiv L_{\psi}(T ) = \left\{f \in L_0 (T ): ||f||_{\psi} := \int_T \psi( |f (x)| ) dx < \infty \right\}.$$ The following problems are investigated: Relationship between the rate of approximation of $f$ by trigonometric polynomials in $L_{\psi}$ and smoothness in $L_1$. Correlation between the moduli of continuity of $f$ in $L_{\psi}$ and $L_1$, and theorems on imbedding of the classes $\text{Lip} (\alpha, \psi)$ in $L_1$. Structure of functions from the class $\text{Lip}(1, \psi)$.

### Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations

Shao Yuan Huang, Sui Sun Cheng

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1233-1248

In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$ It is important to note that the condition $g'(t) \geq \alpha > 0$ is required. In this paper, we remove this restriction under the superlinear assumption $\eta > 0$. Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations.

### Best approximations of periodic functions in generalized lebesgue spaces

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1249-1265

In generalized Lebesgue spaces with variable exponent, we determine the order of the best approximation on the classes of $(\psi, \beta)$-differentiable $2\pi$-periodic functions. We also obtain an analog of the well-known Bernstein inequality for the $(\psi, \beta)$-derivative, with the help of which the converse theorems of approximation theory are proved on the indicated classes.

### Representations of canonical anticommutation relations with orthogonality condition

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1266-1272

We study the class of Hilbert space representations of the ∗-algebra $A^{(d)}_0$ generated by relations of the form $$A^{(d)}_0 = \mathbb{C}\langle a_j, a_j^{*} | a_j^{*} a_j = 1 - a_j a_j^{*},\; a_j, a_j^{*} = 0, i \neq j,\; i, j = 1,...,d\rangle,$$ Namely, we describe the classes of unitary equivalence of irreducible representations of $A^{(d)}_0$ such that there exists $j = 1,...,d$ for which $a^2_j \neq 0$.

### Geodesic spaces tangent to metric spaces

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1273-1281

We investigate the geometry of spaces tangent and pretangent to general metric spaces with marked point. We find a sufficient condition under which every separable tangent space is geodesic. This condition is almost exact in the sense that it is necessarily satisfied if all spaces pretangent to a given metric space are geodesic.

### On one Shemetkov problem

Semenchuk V. N., Velesnitskii V. F.

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1282-1288

This work is devoted to the investigation of the structure of superradical formations.

### Generalized Weyl's theorem and tensor product

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1289-1296

We give necessary and/or sufficient conditions ensuring the passage of generalized a-Weyl theorem and property $(gw)$ from $A$ and $B$ to $A \otimes B$.