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1.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1994,46(5):581-603
We study the minimality of elementsx
h,j,k
of canonical systems of root vectors. These systems correspond to the characteristic numbers
k
of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space
. In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC
v
x
h,j,k
is minimal in
for arbitrary positive <1+, provided that ¦k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology. 相似文献
2.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1991,43(9):1136-1150
On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x(
n
)+A
1
x(n–1 +...+
n
A
n
x, the matrices A1,...,An are of size m × m, x is an unknown and f a given function with values in the m-dimensional space
m
, F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991. 相似文献
3.
Let U
j be a finite system of functionals of the form
, and let
be the subspace of the Sobolev space
, 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
j
for every function
j
, and if
j
=
s
for j s, then k
j k
s. For the K-functional
we establish the inequality
, where the constant c > 0 does not depend on (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. 相似文献
4.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1992,44(7):857-874
The equivalence of derived chains constructed from the principal vectors of polynomial sheafs of operators acting in Hilbert space is studied. These derived chains correspond to different boundary-value problems on the semi-axis for operator-differential equations whose symbol is these operator sheafs. On the basis of equivalence tests assertions are deduced concerning the minimality of derived chains corresponding to a boundary-value problem on the semi-axis in the case in which the initial conditions of the vector solution at zero are known, and the solution itself obeys radiation-type conditions at infinity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 948–970, July, 1992. 相似文献
5.
G. V. Radzievskii 《Functional Analysis and Its Applications》1995,29(3):217-219
Mathematical Institute, National Ukrainian Academy of Science. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 87–90, July–September, 1995. 相似文献
6.
7.
8.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1996,48(11):1739-1757
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 ? ∈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 =τ s forj ≠s, 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 ? ∈L p . We also establish some other estimates for the consideredK-functional. 相似文献
9.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1997,49(6):844-864
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 相似文献
10.
We study the question of the number of linearly independent solutions of the equationy
(n) (x)+(Fy) (x)+n
y (x)=0,x [0, 1], in which F is a bounded linear operator acting on various normed function spaces. A number of assertions about the asymptotic behavior of these solutions with respect to , tending to infinity are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1460–1469, November, 1990. 相似文献