# Volume 68, № 4, 2016

### Polynomial approximation in Bergman spaces

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 435-448

The purpose of this work is to obtain Jackson and converse inequalities of the polynomial approximation in Bergman spaces. Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We proved some simultaneous approximation theorems and obtained the Nikolskii – Stechkin inequality for polynomials in these spaces.

### Approximation of some classes of set-valued periodic functions by generalized trigonometric polynomials

Babenko V. F., Babenko V. V., Polishchuk M. V.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 449-459

We generalize some known results on the best, best linear, and best one-sided approximations by trigonometric polynomials from the classes of $2 \pi$ -periodic functions presented in the form of convolutions to the case of classes of set-valued functions.

### Maximum principle for the Laplacian with respect to the measure in a domain of the Hilbert space

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 460-468

We obtain the maximum principle for two versions of the Laplacian with respect to the measure, namely, for the “classical” and “$L^2$” versions in a domain of the Hilbert space.

### Analogs of the spherical transform on the hyperbolic plane

Vasilyanskaya V. S., Volchkov V. V.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 469-484

We introduce the notion of “$s$”-convolution on the hyperbolic plane $H^2$ and consider its properties. Analogs of the Helgason spherical transform on the spaces of compactly supported distributions in $H^2$ are studied. We prove a Paley –Wiener – Schwartz-type theorem for these transforms.

### A note on degree of approximation by matrix means in generalized Hölder metric

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 485-494

The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means.

### Jacobi-type block matrices corresponding to the two-dimensional moment problem: polynomials of the second kind and Weyl function

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 495-505

We continue our investigations of Jacobi-type symmetric matrices corresponding to the two-dimensional real power moment problem. We introduce polynomials of second kind and the corresponding analog of the Weyl function.

### Sufficient conditions for the existence of the $\upsilon$ -density for zeros of entire function of order zero

Mostova M. R., Zabolotskii N. V.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 506-516

We select the subclasses of zero-order entire functions $f$ for which we present sufficient conditions for the existence of $\upsilon$ -density for zeros of $f$ in terms of the asymptotic behavior of the logarithmic derivative F and regular growth of the Fourier coefficients of $F$.

### Global attractors of impulsive infinite-dimensional systems

Kapustyan O. V., Perestyuk N. A.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 517-528

We study the existence of global attractors in discontinuous infinite-dimensional dynamical systems, which may have trajectories with infinitely many impulsive perturbations. We also select a class of impulsive systems for which the existence of a global attractor is proved for weakly nonlinear parabolic equations.

### On weakly periodic Gibbs measures for the Potts model with external field on the Cayley tree

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 529-541

We study the Potts model with external field on the Cayley tree of order $k \geq 2$. For the antiferromagnetic Potts model with external field and $k \geq 6$ and $q \geq 3$, it is shown that the weakly periodic Gibbs measure, which is not periodic, is not unique. For the Potts model with external field equal to zero, we also study weakly periodic Gibbs measures. It is shown that, under certain conditions, the number of these measures cannot be smaller than $2^q - 2$.

### Stability of versions of the Weyl-type theorems under tensor product

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 542-550

We study the transformation versions of the Weyl-type theorems from operators $T$ and $S$ for their tensor product $T \otimes S$ in the infinite-dimensional space setting.

### Multiple Haar basis and m-term appriximations for functions from the Besov classes. I

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 551-562

We describe the isotropic Besov spaces of functions of several variables in the terms of conditions imposed on the Fourier – Haar coefficients.

### Necessary and sufficient conditions for the invertibility of nonlinear differentiable maps

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 563-576

We establish necessary and sufficient conditions for the invertibility of nonlinear differentiable maps in the case of arbitrary Banach spaces. We establish conditions for the existence and uniqueness of bounded and almost periodic solutions of nonlinear differential and difference equations.