# Radzievskii G. V.

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

Nesterenko A. N., Radzievskii G. V.

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 $L_2$.

### On one extremal problem for numerical series

Radzievskaya E. I., Radzievskii G. V.

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 $1 < p < ∞$.

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

Radzievskaya E. I., Radzievskii G. V.

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.

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

Radzievskaya E. I., Radzievskii G. V.

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 *j* ≠ *s*, then *k* _{j} ≠ *k* _{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*.

### 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.

### 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.

### 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*

### 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 following*K*-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] and*W* _{ p,U } ^{ r } is a subspace of the Sobolev space*W* _{ p } ^{ r } [0, 1], 1≤*p*≤∞, which consists of functions*g* 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 }=τ_{ s } for*j* ≠*s*, then*l* _{ 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 function*f* 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 constant*c*>0 does not depend on δ>0 and ƒ ∈*L* _{ p }. We also establish some other estimates for the considered*K*-functional.

### 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* = λ*x*, *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* _{h} ≤ *n* + *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.

### 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.

### 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 $¦μ_k¦ > ρ$.

### 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.

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

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

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

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

### 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

### 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

### 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

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

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

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

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

### 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

### 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