# Volume 55, № 5, 2003

### Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 579-589

We investigate the relationship between the constants *K*(**R**) and *K*(**T**), where \(K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}\) is the exact constant in the Kolmogorov inequality, **R** is the real axis, **T** is a unit circle, $$L_{p,s}^r (G)$$ is the set of functions *x* ∈ *L* _{p}(*G*) such that *x* ^{(r)} ∈ *L* _{s}(*G*), *q*, *p*, *s* ∈ [1, ∞], *k*, *r* ∈ **N**, *k* < *r*, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$
then*K*(*R*) = *K*(*T*),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$
then*K*(**R**) ≤ *K*(**T**); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis.

### Behavior of the Double-Layer Potential for a Parabolic Equation on a Manifold

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 590-603

We prove that, similarly to the double-layer potential in \(\mathbb{R}^n \) , the double-layer potential constructed in a Riemann manifold of nonpositive sectional curvature has a jump in passing through the surface where its density is defined.

### On the Solution of Problems of Nonlinear Conditional Optimization on Arrangements by the Cut-Off Method

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 604-611

We propose an exact method for the solution of a minimization problem on arrangements of a linear objective function with linear and concave additional constraints. We prove the finiteness of the proposed algorithm of the cut-off method.

### Averaged Synthesis of the Optimal Control for a Wave Equation

Kapustyan O. V., Sukretna A. V.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 612-620

For a wave equation, we determine an optimal control in the feedback form and prove the convergence of the constructed approximate control to the exact one.

### On Conditions for the Applicability of the Lax–Phillips Scattering Scheme to the Investigation of an Abstract Wave Equation

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 621-630

We find necessary and sufficient conditions under which orthogonal incoming and outgoing subspaces exist for a group of solutions of an abstract wave equation and possess an additional property of “equivalence” with respect to the operator of time reversion.

### On One Problem of the Investigation of Global Solutions of Linear Differential Equations with Deviating Argument

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 631-640

We present conditions under which global solutions of linear systems of differential equations with deviating argument are solutions of ordinary differential equations.

### Construction of an Integral Manifold of a Multifrequency Oscillation System with Fixed Times of Pulse Action

Petryshyn R. I., Samoilenko A. M., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 641-662

We determine a class of multifrequency resonance systems with pulse action for which an integral manifold exists. We construct a function that determines a discontinuous integral manifold and investigate its properties.

### Best “Continuous” $n$-Term Approximations in the Spaces $S_\phi ^p$

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 663-670

We find exact values of upper bounds for the best approximations of $q$-ellipsoids by polynomials of degree $n$ in the spaces $S_\phi ^p$ in the case where the approximating polynomials are constructed on the basis of $n$-dimensional subsystems chosen successively from a given orthonormal system ϕ.

### On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 671-686

We describe dual spaces of classes of the Hardy–Sobolev type \(F_l^{pq} (U^n )\) of functions holomorphic in a polydisk for 0 < *p* ≤ 1 and *q* ∈ (0, ∞) and for *p* ∈ (1, ∞) and *q* = 1.

### Smooth and Topological Equivalence of Functions on Surfaces

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 687-700

We obtain conditions under which the Morse functions defined on surfaces are smooth equivalent and functions with isolated critical (singular) points are topologically equivalent.

### Criterion for the Denseness of Algebraic Polynomials in the Spaces $L_p \left( {{\mathbb{R}},d {\mu }} \right)$, $1 ≤ p < ∞$

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 701-705

The criterion for the denseness of polynomials in the space $L_p \left( {{\mathbb{R}},d {\mu }} \right)$ established by Hamburger in 1921 is extended to the spaces $L_p \left( {{\mathbb{R}},d {\mu }} \right)$, $1 ≤ p < ∞$.

### The Jacobi Field of a Lévy Process

Berezansky Yu. M., Lytvynov E. V., Mierzejewski D. A.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 706-710

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( \(\mathbb{R}\) -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.

### On the Application of the Averaging Principle in Stochastic Differential Equations of Hyperbolic Type

Kolomiets O. V., Kolomiyets V. G., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 711-715

We prove a theorem on the application of the Bogolyubov–Mitropol'skii averaging principle to stochastic partial differential equations of the hyperbolic type.

### Construction of Separately Continuous Functions with Given Restriction

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 716-721

We solve the problem of the construction of separately continuous functions on a product of two topological spaces with given restriction. It is shown, in particular, that, for an arbitrary topological space *X* and a function *g*: *X* → **R** of the first Baire class, there exists a separately continuous function *f*: *X* × *X* → **R** such that *f*(*x*, *x*) = *g*(*x*) for every *x* ∈ *X*.