# Volume 48, № 3, 1996

### Optimization of quadratures on classes of functions given by differential operators with Real Spectra

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 291-300

We study the problem of optimization of quadrature formulas for broad classes of periodic functions defined in terms of differential operators with real spectra. We analyze quadrature formulas containing values of functions and values of the images of functions under the action of some differential operators. The rectangular formula is proved to be optimal.

### On exact values of quasiwidths of some classes of functions

Shabozov M. Sh., Vakarchuk S. B.

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 301-308

In the Hilbert space *L* _{2}(Δ^{2}), Δ = [0, 2 π] we establish exact estimates of the Kolmogorov quasiwidths of some classes of periodic functions of two variables whose averaged modules of smoothness of mixed derivatives are majorizable by given functions.

### Generalized moment representations and invariance properties of Padé approximants

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 309-314

By the method of generalized moment representations, we generalize the well-known invariance properties of Padé approximants under linear-fractional transformations of approximated functions.

### Valiron-type and valiron-titchmarsh-type theorems for entire functions of order zero

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 315-325

By using known asymptotics of the counting function of zeros of an entire function *f* of order zero, we determine the asymptotics of In *f* under the condition that all zeros of*f* lie on the same ray. The inverse problem is also analyzed.

### Copositive pointwise approximation

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 326-334

We prove that if a function*f* ∈*C* ^{(1)} (*I*),*I*: = [−1, 1], changes its sign*s* times (*s* ∈ ℕ) within the interval*I*, then, for every*n* > *C*, where*C* is a constant which depends only on the set of points at which the function changes its sign, and*k* ∈ ℕ, there exists an algebraic polynomial*P* _{ n } =*P* _{ n }(*x*) of degree ≤*n* which locally inherits the sign of*f(x)* and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω_{ k } (*f*′;*t*) is the*k*th modulus of continuity of the function*f*’. It is also shown that if*f* ∈*C* (*I*) and*f*(*x*) ≥ 0,*x* ∈*I* then, for any*n* ≥*k* − 1, there exists a polynomial*P* _{ n } =*P* _{ n } (*x*) of degree ≤*n* such that*P* _{ n } (*x*) ≥ 0,*x* ∈*I*, and |*f*(*x*) −*P* _{ n }(*x*)| ≤*c*(*k*)ω_{ k } (*f*;*n* ^{−2} +*n* ^{−1} √1 −*x* ^{2}),*x* ∈*I*.

### Operator methods in the problem of perturbed motion of a rotating body partially filled with liquid

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 335-348

We apply operator methods to the investigation of an initial boundary-value problem which describes the perturbed motion of a body with cavity partially filled with an ideal liquid relative to the uniform rotation of this system about a fixed axis. We prove the existence and uniqueness of generalized solutions with finite energy and establish a sufficient condition for the stability of motion and some properties of the spectrum of the problem under consideration.

### Nonperiodic locally solvable*T*(*Ā*)-groups

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 349-361

We constructively describe nonperiodic locally solvable*T*(Ā) -groups and select three types of groups of this sort.

### Nonlocal parabolic boundary-value problem

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 362-367

For a parabolic system with time-dependent coefficients, we construct its Green function and present the classical solution of a two-point problem with nonlocal boundary conditions given by differential polynomials.

### Tauberian and Abelian Theorems for random fields with strong dependence

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 368-382

We prove Tauberian and Abelian theorems for Hankel-type integral transformations.

### Absolutely decomposable groups

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 383-392

We establish numerous results concerning the construction of decomposable and absolutely decomposable groups.

### Strong summability of orthogonal expansions of summable functions. II

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 393-405

We study the problem of strong summability of Fourier series in orthonormal systems of polynomialtype functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of the systems under consideration.

### Generalized periodic solutions of quasilinear equations

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 406-411

We study a boundary-value periodic problem for the quasilinear equation*u* _{ ff } −*u* _{ xx } =*F*[*u*,*u* _{ f } *u* _{ x }],*u*(0,*t*) =*u* (π,*t*),*u* (*x, t* + π/*q*) =*u*(*x, t*), 0 ≤*x* ≤*π*,*t* ∈ ℝ,*q* ∈ ℕ. We establish conditions under which the theorem on the uniqueness of a smooth solution is true.

### Generalization of the fricke theorem on entire functions of finite index

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 412-417

We prove that, for every sequence (*a* _{k}) of complex numbers satisfying the conditions Σ(1/|*a* _{ k }|) < ∞ and |*a* _{ k+1}| − |*a* _{ k }| ↗ ∞ (*k* → ∞), there exists a continuous function*l* decreasing to 0 on [0, + ∞] and such that *f*(*z*) = Π(1 −*z*/|*a* _{ k }|) is an entire function of finite *l*-index.

### Solvable groups of finite non-abelian sectional rank

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 418-421

We study non-Abelian solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite.

### On the existence of an invariant manifold for weakly nonlinear matrix differential equations

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 422-424

For a system of nonlinear matrix differential equations with small parameter, we establish conditions of the existence of an invariant manifold in terms of the Green function.

### Turnpike theorems for convex problems

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 425-428

We prove turnpike theorems for systems described by differential inclusions with convex graphs.

### On groups factorizable in commuting almost locally normal subgroups

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 429-431

We prove that an *RN*-group (in particular, locally solvable) *G* =*G* _{1} *G* _{2} ...*G* _{ n } with *G* _{ i } and π(*G* _{ i }) ∩ π(*G* _{ j }) = ⊘,*i* ≠*j* is a periodic hyper-Abelian group if the subgroups*G* _{j} are almost locally normal.