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1.
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
2.
Let
be a nondecreasing sequence of positive numbers and let l
1,α be the space of real sequences
for which
. We associate every sequence ξ from l
1,α with a sequence
, where ϕ(·) is a permutation of the natural series such that
, j ∈ ℕ. If p is a bounded seminorm on l
1,α and
, then
Using this equality, we obtain several known statements.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 1002–1006, July, 2005. 相似文献
3.
Let Γ be the set of all permutations of the natural series and let α = {α j}
j∈ℕ, ν = {νj}
j∈ℕ, and η = {ηj}
j∈ℕ be nonnegative number sequences for which
is defined for all γ:= {γ(j)}
j∈ℕ ∈ Γ and η ∈ l
p. We find
in the case where 1 < p < ∞.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1430–1434, October, 2005. 相似文献
4.
We show that the remainder of the Taylor expansion for a holomorphic function can be written down in Lagrange form, provided that the argument of the function is sufficiently close to the interpolation point. Moreover, the value of the derivative in the remainder can be taken in the intersection of the disk whose diameter joins the interpolation point and the argument of the function and an arbitrary small angle whose bisectrix is the ray from the interpolation point through the argument of the function. 相似文献
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