# Volume 55, № 7, 2003

### Stable Bundles on a Rational Curve with One Simple Double Point

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 867-874

We give a classification of stable vector bundles on a rational curve with one simple double point.

### On the Inversion of the Local Pompeiu Transformation

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 875-880

The inversion of the local Pompeiu transform for a triangle is constructed.

### Existence Theorems for Generalized Moment Representations

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 881-888

We establish conditions for the existence of generalized moment representations introduced by Dzyadyk in 1981.

### Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 889-900

For a sectorial operator *A* with spectrum σ(*A*) that acts in a complex Banach space *B*, we prove that the condition σ(*A*) ∩ *i* **R** = Ø is sufficient for the differential equation \(\varepsilon x_\varepsilon^\prime\prime(t)+x_\varepsilon^\prime(t)=Ax_\varepsilon(t)+f(t), t \in R,\) where ε is a small positive parameter, to have a unique bounded solution *x* _{ε} for an arbitrary bounded function *f*: **R** → *B* that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on **R** as ε → 0+ to the unique bounded solution of the differential equation *x*′(*t*) = *Ax*(*t*) + *f*(*t*).

### Inverse Problem with Free Boundary for Heat Equation

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 901-910

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a one-dimensional heat equation with unknown time-dependent leading coefficient in the case where a part of the boundary of the domain is unknown.

### Multiple Fourier Sums on Sets of $\bar \psi$ -Differentiable Functions (Low Smoothness)

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 911-918

We investigate the behavior of deviations of rectangular partial Fourier sums on sets of $\bar \psi$-differentiable functions of many variables.

### Best Linear Methods of Approximation of Functions of the Hardy Class $H_p$

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 919-925

We determine the exact value of the best linear polynomial approximation of a unit ball of the Hardy space $H_p, 1 ≤ p ≤ ∞$, on concentric circles $Tρ = z ∈ C:|z|=ρ, 0 ≤ ρ < 1$, in the uniform metric. We construct the best linear method of approximation and prove the uniqueness of this method.

### On One Sequence of Polynomials and the Radius of Convergence of Its Poisson–Abel Sum

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 926-934

For one sequence of polynomials arising in the construction of the numerical-analytic method for finding periodic solutions of nonlinear differential equations, we determine the explicit form of the Poisson–Abel sum and the exact solution of the equation for finding the radius of convergence of this sum.

### Effect of Time Delay of Support Propagation in Equations of Thin Films

Shishkov A. E., Taranets R. M.

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 935-952

We prove the existence of the effect of time delay of propagation of the support of “strong” solutions of the Cauchy problem for an equation of thin films and establish exact conditions on the behavior of an initial function near the free boundary that guarantee the appearance of this effect.

### Analysis of the Accuracy of Interpolation of Entire Operators in a Hilbert Space in the Case of Perturbed Nodal Values

Kashpur O. F., Khlobystov V. V.

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 953-960

In a Hilbert space with Gaussian measure, we obtain an estimate for the accuracy of interpolation of an entire operator in the case where its values are perturbed at nodes and determine the value of the degree of an interpolation polynomial the exceeding of which does not improve the estimate of the accuracy of interpolation.

### Construction of Asymptotic Solutions of Linear Systems of Differential Equations with Two Small Parameters

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 961-976

We consider a linear homogeneous system of differential equations with two small parameters. In this system, the dependence on one parameter is regular and on the other is singular. Using methods of the theory of perturbations of linear operators and a space analog of the Newton diagrams, we investigate the asymptotics of a general solution of this system in the case where its leading matrix has a multiple eigenvalue associated with a multiple elementary divisor.

### Perturbed Parabolic Equation on a Riemannian Manifold

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 977-982

We construct a fundamental solution of an equation with perturbed diffusion operator on a manifold of nonnegative curvature.

### Extension of a Function from the Exterior of an Interval to a Positive-Definite Function on the Entire Axis and an Approximation Characteristic of the Class $W_M^{r, β}$

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 983-990

We establish sufficient conditions for the extension of a function defined on $[a, +∞)$, where $a > 0$, to a positive-definite function on the entire axis.

### Discrepancy Principle and Convergence Rates in Regularization of Monotone Ill-Posed Problems

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 991-997

The convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear monotone ill-posed problems in a Banach space are established on the basis of the choice of a regularization parameter by the Morozov discrepancy principle.

### On Bicyclic *T*-Factorizability in the Class *T*[14, 6]

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 998-1005

We completely solve the problem of the existence of *T*-factorizations in the class of trees of order 14 with the largest vertex order 6.

### Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 1006-1009

For the equation *L* _{0} *x*(*t*) + *L* _{1} *x* ^{(1)}(*t*) + ... + *L* _{n} *x* ^{(n)}(*t*) = 0, where *L* _{k}, *k* = 0, 1, ... , *n*, are operators acting in a Banach space, we formulate conditions under which a solution *x*(*t*) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.