# Volume 64, № 8, 2012

### Approximation of some classes of functions of many variables by harmonic splines

Babenko V. F., Leskevich T. Yu.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1011-1024

We determine the exact values of upper bounds of the error of approximation by harmonic splines for functions $u$ defined on an $n$-dimensional parallelepiped $\Omega$ forwhich $||\Delta u||_{L_{\infty}(\Omega)} \leq 1$ and for functions $u$ defined on $\Omega$ forwhich $||\Delta u||_{L_{p}(\Omega)} \leq 1, \quad 1 \leq p \leq \infty$. In the first case, the error is estimated in $L_{p}(\Omega), \quad 1 \leq p \leq \infty$; in the second case, it is estimated in $L_{1}(\Omega)$.

### On the best polynomial approximation in the space L2 and widths of some classes of functions

Vakarchuk S. B., Zabutnaya V. I.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1025-1032

We consider the problem of the best polynomial approximation of $2\pi$-periodic functions in the space $L_2$ in the case where the error of approximation $E_{n-1}(f)$ is estimated in terms of the $k$th-order modulus of continuity $\Omega_k(f)$ in which the Steklov operator $S_h f$ is used instead of the operator of translation $T_h f (x) = f(x + h)$. For the classes of functions defined using the indicated smoothness characteristic, we determine the exact values of different $n$-widths.

### Derived categories of nodal curves

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1033-1040

We describe derived categories of coherent sheaves over nodal noncommutative curves of string and almost string types.

### Trigonometric widths of classes of periodic functions of many variables

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1041-1052

We obtain exact-order estimates for the trigonometric widths of the classes $B^{\Omega}_{p\theta}$ of periodic functions of many variables in the space $L_q$ for some relations between the parameters $p$ and $q$.

### Nevanlinna formula for the truncated matrix trigonometric moment problem

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1053-1066

This paper is a continuation of our investigation on the truncated matrix trigonometric moment problem begun in Ukr. Mat. Zh. - 2011. - 63, № 6. - P. 786-797. In the present paper, we obtain the Nevanlinna formula for this moment problem in the general case. We assume here that there is more than one moment and the moment problem is solvable and has more than one solution. The coefficients of the corresponding matrix linear fractional transformation are expressed in explicit form via prescribed moments. Simple determinacy conditions for the moment problem are presented.

### Space-time fractional Cauchy problem in spaces of generalized functions

Lopushanskaya G. P., Lopushanskyi A. O.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1067-1079

We prove a theorem on the existence and uniqueness and obtain a representation using the Green vector function for the solution of the Cauchy problem $$u^{(\beta)}_t + a^2(-\Delta)^{\alpha/2}u = F(x, t), \quad (x, t) \in \mathbb{R} ^n \times (0, T], \quad a = \text{const} $$ $$u(x, 0) = u_0(x), \quad x \in \mathbb{R} ^n$$ where $u^{(\beta)}_t$ is the Riemann-Liouville fractional derivative of order $\beta \in (0,1)$, and $u_0$ and $F$ belong to some spaces of generalized functions. We also establish the character of the singularity of the solution at $t = 0$ and its dependence on the order of singularity of the given generalized function in the initial condition and the character of the power singularities of the function on right-hand side of the equation. Here, the fractional $n$-dimensional Laplace operator $\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.

### Periodic solutions of a parabolic equation with homogeneous Dirichlet boundary condition and linearly increasing discontinuous nonlinearity

Fedyashev M. S., Pavlenko V. N.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1080-1088

We consider a resonance problem of the existence of periodic solutions of parabolic equations with discontinuous nonli-nearities and a homogeneous Dirichlet boundary condition. It is assumed that the coefficients of the differential operator do not depend on time, and the growth of the nonlinearity at infinity is linear. The operator formulation of the problem reduces it to the problem of the existence of a fixed point of a convex compact mapping. A theorem on the existence of generalized and strong periodic solutions is proved.

### Asymptotic *m*-phase soliton-type solutions of a singularly perturbed Korteweg?de Vries equation with variable coefficients. II

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1089-1105

We consider the problem of the construction of higher terms of asymptotic many-phase soliton-type solutions of the singular perturbed Korteweg – de Vries equation with variable coefficients. The accuracy with which the obtained asymptotic solution satisfies the original equation is determined.

### Estimates for bilinear approximations of the classes TeX of periodic functions of two variables

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1106-1120

We obtain exact-order estimates for the best bilinear approximations of the classes $S_{p, \theta}^{\Omega} B$ of periodic functions of two variables in the space $L_q$ for some relations between the parameters $p, q, \theta$.

### Generalized relaxed elastic line on an oriented surface

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1121-1131

We study the relaxed elastic line in a more general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for the generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfaces.

### One improvement of the law of the iterated logarithm for the maximum scheme

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1132-1137

A lower bound is found in the law of the iterated logarithm for the maximum scheme.

### An admissible estimator for the *r* th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function

Alikhani S., Mahmoudi E., Torabi H.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1138-1147

We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in the family of transformed chi-square distributions is also given.

### Approximation of holomorphic functions of Zygmund class by Fejer means

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1148-1152

We obtain an asymptotic equality for the upper bounds of deviations of Fejer means on the Zygmund class of functions holomorphic in the unit disk.