The elements of E(Ω)(RN) are called Ω-ultradifferentiable functions of Beurling type. <...> If ρ is surjective, we will say that a version of Borel’s extension theorem holds for the space E(Ω)(RN) (for the original Borel’s extension theorem see [7]). <...> For minimal Beurling class (ωn = nω, ω is nonquasianalytic and ω(2t) = O(ω(t)) as t → ∞) Meise and Taylor [8] have shown that E(Ω)(RN) admits a version of Borel’s extension theorem if and only if ω is strong, i. e. there exists a C > 0 such that almost subadditive weight function, has been studied by the author in [4]. <...> First recall that condition (I) for each n ∈ N there exist m ∈ N and C > 0 such that Pωn(z) ωm(|z|)+C for all z ∈ C provides that Borel’s extension theorem holds for the corresponding class E(Ω)(RN) of minimal or normal type independently of the number N of variables. implies (I), and so (I). <...> Thus, the operator ρ : E(Ω)(R) → E 1 conditions of [8] and [4] for spaces of UDF of minimal and normal type. <...> It was shown in [8] that space of UDF of minimal type defined by this function admits the analog of Borel’s extension theorem. <...> Владикавказский математический журнал январь–март, 2007, Том 9, Выпуск 1 UDC 517.98 REPRESENTATION AND EXTENSION OF ORTHOREGULAR BILINEAR OPERATORS Buskes G., Kusraev A. G.1 In this paper we study some important structural properties of orthosymmetric bilinear operators using the concept of the square of an Archimedean vector lattice. <...> Some new results on extension and analytical representation of such operators are presented. <...> Mathematics Subject Classification (1991): 46A40, 47A65 Key words: vector lattice, positive bilinear operator, orthosymmetric bilinear operator, orthoregular bilinear operator, lattice bimorphism. <...> Introduction Recently the class of orthosymmetric bilinear operators in vector lattices, introduced in [12], has aroused considerable interest. <...> For example, a positive orthosymmetric bilinear operator is symmetric [12] and every positive orthosymmetric bilinear operator defined on a sublattice of an f-algebra can be factored through a positive linear operator and the algebra multiplication <...>
Владикавказский_математический_журнал_№1_2007.pdf
Р О С С И Й С К А Я А К А Д Е М И Я Н А У К
В Л А Д И К А В К А З С К И Й Н А У Ч Н Ы Й Ц Е Н Т Р
ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ И ИНФОРМАТИКИ
ВЛАДИКАВКАЗСКИЙ
МАТЕМАТИЧЕСКИЙ ЖУРНАЛ
Том 9, Выпуск 1
январь–март, 2007
СОДЕРЖАНИЕ
Abanina D. A. On Borel’s extension theorem for general Beurling classes of
ultradifferentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Buskes G., Kusraev A. G. Representation and extension of orthoregular
bilinear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Danchev P. V. A note on weakly ℵ1-separable p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Grabarnik G. Ya., Katz A. A., Shwartz L. On non-commutative ergodic
type theorems for free finitely generated semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Коробейник Ю. Ф. О нулях одного класса гармонических функций . . . . . . . . . . . 48
Teamah A. A. M., Bakouch H. S. Some asymptotic properties of a kernel
spectrum estimate with different multitapers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Тюриков Е. В. Об одной граничной задаче теории бесконечно малых
изгибаний поверхности . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Владикавказ
2007
Стр.1
Владикавказский математический журнал
январь–март, 2007, Том 9, Выпуск 1
UDC 517.51+517.98
ON BOREL’S EXTENSION THEOREM FOR GENERAL BEURLING
CLASSES OF ULTRADIFFERENTIABLE FUNCTIONS
Abanina D. A.
We obtain necessary and sufficient conditions under which general Beurling class of ultradifferentiable
functions admits a version of Borel’s extension theorem.
Mathematics Subject Classification (2000): 46E10,26E10,30D15.
Key words: Ultradifferentiable functions, Borel’s extension theorem.
1. Introduction
function if
Definition 1.1. An increasing continuous function ω : [0,∞)→[0,∞) is called a weight
log t = o(ω(t)), t→∞;
ω(t) = O(t); t→∞;
ϕω(x) := ω(ex) is convex on [x0,∞).
A weight function ω with
∞
property: for each n ∈ N there exists a Cn > 0 such that
ωn(t)+log(t+1) ωn+1(t)+Cn for t 0.
Denote byW↑ the set of all sequences Ω = {ωn}∞
1
ByWnq
Without loss of generality we can assume that
ωn(t) ωn+1(t) for t 0 and n ∈ N.
↑ denote the set of all sequences Ω = {ωn}∞
The Young conjugate ϕ∗
ω : [0,∞)→[0,∞) of ϕω is defined by
ϕ∗
ω(y) := sup{xy −ϕω(x) : x 0}.
For A ∈ (0,∞) we define the space
Eω(ΠN
A) :=
where ΠN
f ∈ C∞(ΠN
A) : |f|ω,A,N := sup
α∈NN
RN, |α| := α1 +. . .+αN for α = (α1, . . . ,αN) ∈ NN
c
2007 D. A. Abanina
0 xA eϕ∗
sup
0 , f(α) :=
f(α)(x)
ω(|α|) <∞ ,
A := {x ∈ RN : x A}, x := max{|xj| : 1 j N} for x = (x1, . . . ,xN) ∈
∂|α|f
∂xα1
1 . . . ∂xαN
N
.
t−2ω(t) dt <∞ is called nonquasianalytic.
n=1 of weight functions with the folllowing
(1)
n=1 of nonquasianalytic weight functions ωn.
Стр.3