2019
Том 71
№ 10

### All Issues

Articles: 21
Article (Russian)

### Multiparameter inverse problem of approximation by functions with given supports

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1116–1127

Let $L_p(S),\;0 < p < +∞$, be a Lebesgue space of measurable functions on $S$ with ordinary quasinorm $∥·∥_p$. For a system of sets $\{B t |t ∈ [0, +∞)^n \}$ and a given function $ψ: [0, +∞) n ↦ [ 0, +∞)$, we establish necessary and sufficient conditions for the existence of a function $f ∈ L_p(S)$ such that $\inf \{∥f − g∥^p_p| g ∈ L_p(S),\;g = 0$ almost everywhere on $S\B t } = ψ (t), t ∈ [0, +∞)^n$. As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in $L_2$.

Brief Communications (Russian)

### On one extremal problem for numerical series

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1430–1434

Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\| {\nu (\alpha \eta )_\gamma } \right\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{\eta :\left\| \eta \right\|_p = 1} \inf _{\gamma \in \Gamma } \left\| {\nu (\alpha \eta )_\gamma } \right\|_1$ in the case where $1 < p < ∞$.

Brief Communications (Ukrainian)

### On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 1002–1006

Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$ is a permutation of the natural series such that $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$. If $p$ is a bounded seminorm on $l_{1,α}$ and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then $$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$ Using this equality, we obtain several known statements.

Article (Russian)

### Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1530-1540

Let U j be a finite system of functionals of the form $U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)$ , and let $W_{p,U}^r$ be the subspace of the Sobolev space $W_p^r [0;1]$ , 1 ≤ p ≤ +∞, that consists only of functions g such that U j(g) = 0 for k j < r. It is assumed that there exists at least one jump τ j for every function σ j , and if τ j = τ s for js, then k jk s. For the K-functional $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ we establish the inequality $K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)$ , where the constant c > 0 does not depend on δ ε (0; 1], the functions f belong to L p, and r = 1, ¨, n. On the basis of this inequality, we also obtain estimates for the K-functional in terms of the modulus of smoothness of a function f.

Article (Ukrainian)

### 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.

Article (Russian)

### On an upper bound for the number of characteristic values of an operator function

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 211–224

We prove a theorem on an upper bound for the number of characteristic values of an operator-valued function that is holomorphic and bounded in a domain. This estimate is similar to the well-known inequality for zeros of a number function that is holomorphic and bounded in a domain. We derive several corollaries of the theorem proved, in particular, a statement on an estimate of the number of characteristic values of polynomial bundles of operators that lie in a given disk.

Article (Russian)

### On the best approximations and rate of convergence of decompositions in the root vectors of an operator

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 754–773

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f

Article (Russian)

### Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1537-1554

We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈L p :=L p [0, 1] andW p,U r is a subspace of the Sobolev spaceW p r [0, 1], 1≤p≤∞, which consists of functionsg such that $\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n}$ . Assume that 0≤l l ≤...≤l n r-1 and there is at least one point τ j of jump for each function σ j , and if τ j s forjs, thenl j l s . Let $\hat f(t) = f(t)$ , 0≤t≤1, let $\hat f(t) = 0$ ,t<0, and let the modulus of continuity of the functionf be given by the equality $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$

We obtain the estimates $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p$ and $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p$ , where β=(pl l + 1)/p(l 1 + 1), and the constantc>0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional.

Article (Russian)

### Asymptotics of eigenvalues of A regular boundary-value problem

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 483-519

We study a boundary-value problem x (n) + Fx = λx, U h(x) = 0, h = 1,..., n, where functions x are given on the interval [0, 1], a linear continuous operator F acts from a Hölder space H y into a Sobolev space W 1 n+s , U h are linear continuous functional defined in the space $H^{k_h }$ , and k hn + s - 1 are nonnegative integers. We introduce a concept of k-regular-boundary conditions U h(x)=0, h = 1, ..., n and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type: $\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n$ , v = ± N, ± N ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and c ± depend on boundary conditions.

Article (Russian)

### Asymptotics of the fundamental system of solutions of a linear functional-differential equation with respect to a parameter

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 811–836

We study a functional-differential equation, where F is a linear operator acting from the Hölder space Hγ into the Sobolev space W p s [0, 1] and ρ is a complex parameter. For large absolute values of ρ, we construct a one-to-one correspondence between the solutions x(ρ;t) and y(ρ;t) of the equations and y(n)+ρyn=0. We also establish conditions that should be imposed on the operatorF in order that specially selected fundamental systems of solutions of these equationsx j (ρ;t) andy j (ρ;t), j=1,...,n, satisfy the estimate with constantsc, κ>0 for the functional space.

Article (Russian)

### 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 $¦μ_k¦ > ρ$.

Article (Russian)

### Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

Ukr. Mat. Zh. - 1994. - 46, № 3. - pp. 279–292

For the equation $L_0x(t)+L_1x′(t) + ... + L_nx^{(n)}(t) = O$, where $L_k, k = 0,1,...,n$, are operators acting in a Banach space, we establish criteria for an arbitrary solution $x(t)$ to be zero provided that the following conditions are satisfied: $x^{(1−1)} (a) = 0, 1 = 1, ..., p$, and $x^{(1−1)} (b) = 0, 1 = 1,...,q$, for $-∞ < a < b < ∞$ (the case of a finite segment) or $x^{(1−1)} (a) = 0, 1 = 1,...,p,$ under the assumption that a solution $x(t)$ is summable on the semiaxis $t ≥ a$ with its first $n$ derivatives.

Article (Ukrainian)

### Properties of solutions of linear functional-differential equations depending on a parameter

Ukr. Mat. Zh. - 1991. - 43, № 9. - pp. 1213–1231

Article (Ukrainian)

### Minimality of derivative chains, corresponding to a boundary value problem on a finite segment

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 195-205

Article (Ukrainian)

### Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval, for polynomial operator pencils

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 83-95

Article (Ukrainian)

### Multiple minimality of the root vectors of a polynomial operator pencil perturbed by an analytic outside a disc operator-function S (λ) with S (∞) = 0

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 599-610

Article (Ukrainian)

### Linear independence and the completeness of derived chains, corresponding to a boundary-value problem on a finite segment

Ukr. Mat. Zh. - 1987. - 39, № 3. - pp. 358–364

Article (Ukrainian)

### Completeness of derived chains corresponding to boundary problems on a semiaxis

Ukr. Mat. Zh. - 1979. - 31, № 4. - pp. 407–416

Article (Ukrainian)

### Completeness of derived chains that correspond to boundary-value problems on a finite segment

Ukr. Mat. Zh. - 1979. - 31, № 3. - pp. 279–288

Article (Ukrainian)

### On the problem of completeness of root vectors corresponding to the two spectral series of sheaves of M. V. Keldysh

Ukr. Mat. Zh. - 1976. - 28, № 3. - pp. 413–418

Article (Ukrainian)

### Certain tests for the degree of completeness of the root vectors of operator functions analytic in a sector

Ukr. Mat. Zh. - 1976. - 28, № 2. - pp. 203–212