# Volume 46, № 5, 1994

### On the uniqueness of elements of the best approximation and the best one-sided approximation in the space $L_1$

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 475–483

The problem of uniqueness of the best approximations in the space $L_1[a, b]$ is studied We consider the problem of the best approximation and the best $(\alpha, \beta)$-approximation of continuous functions and the problem of the best one-sided approximation of continuously differentiable functions.

### On zeros of functions analytic in a half plane and completeness of systems of exponents

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 484–500

Sequences of zeros are described for functions $f$ analytic in the right halfplanc and satisfying the condition $|f(z)| \leq 0(1) \exp (\sigma|z|),\quad 0 \leq \sigma \leq \infty$ criterion of completeness of a system of exponentials in a space of functions analytic in a semistrip is established.

### On the upper bound of uniformly bounded solutions

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 501–505

New conditions of uniform boundedness of solutions are obtained and methods for calculating upper bounds are suggested.

### Testing hypotheses by using optimal statistical criteria. II

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 506–515

We study the structure of the critical function of an optimal statistical criterion for testing an arbitrary finite set of simple alternative hypotheses.

### Coordinate system and combinatorial objects (an example of a generalized quadrangle)

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 516–523

By using a generalized quadrangle as an example, we verify the assumption that coordinate systems (different from the standard Cartesian coordinate system) exist not only in an arbitrary projective plane, where they are determined by a nondegenerate quadrangle, but also in some other combinatorial objects.

### Three-dimensional initial boundary-value problem of the convection of a viscous weakly compressible fluid. I. Global solvability

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 524–536

By using asymptotic methods, we study the three-dimensional initial boundary-value problem of the convection of a viscous thermally inhomogeneous weakly compressible fluid which fills a cavity in a solid body. A theorem about the global solvability of this problem (with respect to time) is proved. For solving this problem, we suggest a convergent iteration process of a special form.

### Reduction and coisotropic invariant tori of Hamiltonian systems with non-poisson commutative symmetries. I

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 537–544

Hamiltonian systems invariant under the non-Poisson torus action are studied on a symplectic manifold. Conditions are established under which coisotropic invariant tori filled with quasiperiodic motions exist in these systems.

### Minimality of root vectors of operator functions analytic in an angle

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 545–566

We study the minimality of elements $x_{h, j, k}$ of canonical systems of root vectors. These systems correspond to the characteristic numbers $μ_k$ of operator functions $L(λ)$ analytic in an angle; we assume that operators act in a Hilbert space $H$. In particular, we consider the case where $L(λ) = I + T(λ)C^{β} > 0, \;I$ is an identity operator, $C$ is a completely continuous operator, $∥(I- λC)^{−1}∥ ≤ c$ for $|\arg λ| ≥ θ,\; 0 < θ < π$, the operator function $T(λ)$ is analytic, and $T(λ)$ for $|\arg λ| < θ$. It is proved that, in this case, there exists $ρ > 0$ such that the system of vectors $C^v_{x_{h,j,k}}$ is minimal in $ H$ for arbitrary positive $ν < 1+β,$ provided that $¦μ_k¦ > ρ$.

### Existence of a multiplicative basis for a finitely spaced module over an aggregate

Roiter A. V., Sergeychuk V. V.

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 567–579

It is proved that a finitely spaced module over $k$-category admits a multiplicative basis (such a module gives rise to a matrix problem in which the allowed column transformations are determined by a module structure, the row transformations are arbitrary, and the number of canonical matrices is finite).

### On the existence of piecewise-continuous separatrix curves for a certain class of pulse systems

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 580–585

Sufficient conditions of existence of piecewise continuous separatrix curves are obtained for a certain class of systems of differential equations with pulse influence for small $\varepsilon \geq 0$.

### Filtration formulas for solutions of nonlinear equations with random right-hand sides

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 586–596

Explicit filtration formulas are obtained for the solutions of nonlinear differential equations with random right-hand sides. In the case of a Gaussian random process, these formulas are simplified.

### Approximations in spaces of locally integrable functions

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 597–625

We study approximations of functions from the sets $\hat{L}^{\psi}_{\beta}\mathfrak{N}$, which are determined by convolutions of the following form: $$f(x) = A_0 + \int\limits_{-\infty}^{+\infty}\varphi(x + t) \hat{\psi}_{\beta}(f)dt, \quad \varphi \in \mathfrak{N},\quad \hat{\psi}_{\beta} \in L(-\infty, +\infty)$$ where $\mathfrak{N}$ is a fixed subset of functions with locally integrable $p$-th powers $(p \geq 1)$. As an approximating aggregate, we use so-called Fourier operators, which are entire functions of the exponential type $\leq \sigma$ that turn into trigonometric polynomials if the function $\varphi(\cdot)$ is periodic (in particular, they may be the Fourier sums of the function approximated). Approximations are studied in the spaces $\hat{L}_p$ determined by a locally integrable norm $||\cdot||_{\hat{p}}$. Analogs of the Lebesgue and Favard inequalities, well-known in the periodic case, are obtained and used for finding order-exact estimates of the corresponding best approximations and estimates of approximations by Fourier operators, which are order-exact and, in some important cases, they arc also exact in the sense of constants with principal terms of these estimates.

### Estimation of the moduli of continuity of one-variable functions in the metric of $L$ in terms of fourier coefficients

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 626–632

This paper is a survey of results concerning the estimation of the moduli of continuity of functions in the metric of *L* in terms of their Fourier coefficients. Upper bounds, lower bounds, and asymptotic estimates of the moduli of continuity are presented.

### On the asymptotic behavior of sequences given by recursion relations in a Banach space

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 633–641

The criteria of boundedness and asymptotic periodicity are obtained for certain recursion sequences in a Banach space.

### Estimation of eigenvalues of self-adjoint operators

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 642–648

Estimates of the number of eigenvalues are obtained for perturbations of certain selfadjoint and unitary operators in a Hilbert space. As a special case, we consider a perturbation of the operator of multiplication by an independent variable in $L_2(\mathbb{R})$ and $L_2(0, 1)$.