2018
Том 70
№ 6

# Volume 61, № 4, 2009

Article (Ukrainian)

### Comonotone approximation of twice differentiable periodic functions

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 435-451

In the case where a $2π$-periodic function $f$ is twice continuously differentiable on the real axis $ℝ$ and changes its monotonicity at different fixed points $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ$(i.e., on $ℝ$, there exists a set $Y := {y_i } i∈ℤ$ of points $y_i = y_{i+2s} + 2π$ such that the function $f$ does not decrease on $[y_i , y_{i−1}]$ if $i$ is odd and does not increase if $i$ is even), for any natural $k$ and $n, n ≥ N(Y, k) = const$, we construct a trigonometric polynomial $T_n$ of order $≤n$ that changes its monotonicity at the same points $y_i ∈ Y$ as $f$ and is such that $$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$ $$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/ n),f ∈ C^{(r)},\; r ≥ 2),$$ where $N(Y, k)$ depends only on $Y$ and $k, c(k, s)$ is a constant depending only on $k$ and $s, ω k (f, ⋅)$ is the modulus of smoothness of order $k$ for the function $f$, and $‖⋅‖$ is the max-norm.

Article (Ukrainian)

### $A_2$-continued fraction representation of real numbers and its geometry

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 452-463

We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreover, for $α_1 α_2 = 1/2$, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose $A_2$-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its $A_2$-continued fraction representation form a homogeneous Markov chain are also investigated.

Article (Russian)

### On the theory of stability of matrix differential equations

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 464-471

We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.

Article (Ukrainian)

### Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 472-493

In weighted C-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2m in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.

Article (Russian)

### Approximative characteristics of the isotropic classes of periodic functions of many variables

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 513-523

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov $(B_{p,θ}^r)$ and Nukol’skii $(H_p^r )$ classes of periodic functions of many variables in the metric of $L_q , 1 ≤ p, q ≤ ∞$. We also establish the orders of the best approximations of functions from the same classes in the spaces $L_1$ and $L_{∞}$ by trigonometric polynomials with the corresponding spectrum.

Article (Russian)

### Approximation of the classes $C_{β}^{ψ} H_{ω}$ by generalized Zygmund sums

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 524-537

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set $H_{ω}$ in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.

Article (English)

### On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 538-547

Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ.

Article (Russian)

### Weakly nonlinear boundary-value problem in a special critical case

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 548-562

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case.

Brief Communications (Russian)

### Negative result in pointwise 3-convex polynomial approximation

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 563-567

Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, for an arbitrary algebraic polynomial $P ∈ Δ^3 [−1, 1]$, there exists $x$ such that $$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$ where $C > 0$ is a constant that depends only on $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$.

Brief Communications (Ukrainian)

### Simple derivations of higher degree in two variables

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 568-571

We present a new class of simple derivations of arbitrary degree in the ring of polynomials in two variables.

Brief Communications (Ukrainian)

### Best cubature formulas for some classes of continuous functions of two variables

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 572-576

The exact value of the error of a cubature formula is determined for some classes of continuous functions of two variables defined by strictly monotone moduli of continuity.