# Volume 60, № 6, 2008

### On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 723–736

In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabolic equation with,
generally speaking, discontinuous coefficients. The matrix of senior coefficients satisfies the parabolic Cordes condition with respect to space variables.
We prove that the generalized solution to the problem belongs to the Holder space *C*^{ 1+λ} if the right-hand side *f* belongs to *L _{p}*,

*p*>

*n*.

### Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 737–758

We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the generalized stochastic derivative of a solution of a stochastic equation with Wick-type nonlinearity.

### On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II

Kozachenko Yu. V., Perestyuk M. M.

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 759–775

We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval.

### On the theory of stability of motion of a nonlinear system on a time scale

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 776–782

We investigate the problem of stability of a nonlinear system on a time scale and propose a unified approach to the analysis of stability of motion based on a generalized direct Lyapunov method.

### Linear methods for approximation of some classes of holomorphic functions from the Bergman space

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 783–795

We construct a linear method of the approximation $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ in the unit disk of classes of holomorphic functions $A^{\psi}_p$ that are the Hadamard convolutions of unit balls of the Bergman space $A_p$ with reproducing kernels $\psi(z) = \sum^\infty_{k=0}\psi_k z^k.$ We give conditions on $\psi$ under which the method $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ approximate the class $A^{\psi}_p$ in metrics of the Hardy space $H_s$ and Bergman space $A_s,\; 1 \leq s \leq p,$ with error that coincides in order with a value of the best approximation by algebraic polynomials.

### Energy interaction between linear and nonlinear oscillators (energy transfer through the subsystems in a hybrid system)

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 796–814

The analysis of the energy transfer between subsystems coupled in a hybrid system is an urgent problem for various applications. We present an analytic investigation of the energy transfer between linear and nonlinear oscillators for the case of free vibrations when the oscillators are statically or dynamically connected into a double-oscillator system and regarded as two new hybrid systems, each with two degrees of freedom. The analytic analysis shows that the elastic connection between the oscillators leads to the appearance of a two-frequency-like mode of the time function and that the energy transfer between the subsystems indeed exists. In addition, the dynamical linear constraint between the oscillators, each with one degree of freedom, coupled into the hybrid system changes the dynamics from single-frequency modes into two-frequency-like modes. The dynamical constraint, as a connection between the subsystems, is realized by a rolling element with inertial properties. In this case, the analytic analysis of the energy transfer between linear and nonlinear oscillators for free vibrations is also performed. The two Lyapunov exponents corresponding to each of the two eigenmodes are expressed via the energy of the corresponding eigentime components.

### Saturation of the linear methods of summation of Fourier series in the spaces *S*^{ p}_{φ}

^{ p}

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 815 – 828

We consider the problem of the saturation, in the spaces *S ^{ p}*

_{φ}, of linear summation methods for Fourier series, which are determined by the sequences of functions defined on a subset of the space

*C*. We obtain sufficient conditions for the saturation of such methods in these spaces.

### Combinatorial aspects of the topological classification of functions on a circle

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 829–836

We prove a necessary and sufficient condition of topological equivalence of smooth functions which are given on a circle and have a finite number of local extrema.

### On Kolmogorov-type inequalities for fractional derivatives of functions of two variables

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 837–842

We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order.

### On the conditions of convergence for one class of methods used for the solution of ill-posed problems

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 843–850

We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional conditions for these methods guaranteeing the convergence with a given rate to normal solutions from a certain set.

### On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Mulyava O. M., Sheremeta M. M.

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 851–856

For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence $\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$ It is proved that the condition $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$, is necessary and sufficient for equivalence of relations $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty$ and $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0,$ for each such series.

### Majorants of functions with vanishing integrals over balls

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 857–861

We prove the existence of nontrivial functions in *R ^{n }*,

*n*> 2, with vanishing integrals over balls of fixed radius and given majoranta of growth.

### Besicovitch-Danzer-type characterization of a circle

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 862–864

We investigate a Besicovitch-Danzer-type characterization of a circle in a class of compact sets whose boundary divides the plane into several components.