General function spaces. III(SpacesB p,q g(x) andF p,q g(x) , 1<p<∞: basic properties)

qNАстОьЩАь РАБОтА (А тАк жЕ ЕЕ пРОДОлжЕНИЕ — РА БОтА «ОБЩИЕ ФУНкцИОНАльН ыЕ пРО-стРАНстВА, IV») пОсВьЩЕНА ИсслЕДОВ АНИУ БАНАхОВых пРОст РАНстВB _{p, q} ^g(x) ИF _p,q ^g(x) РАспРЕДЕлЕНИИ (ОБОБЩЕННых) ВR _n . В спЕц ИАльНых слУЧАьх ЁтИ пРОстРАНстВА РОДстВ ЕННы ИжВЕстНыМ клАсс АМ сОБОлЕВА—лЕБЕгА—БЕ сОВА, ИжОтРОпНыМ И АНИ жОтРОпНыМ. жДЕсь РАссМАтРИВАУт сь слЕДУУ-ЩИЕ сВОИстВА: плОтНОсть г лАДкИх ФУНкцИИ, ЁкВИВ АлЕНтНыЕ НОРМы, ИНтЕРпОльцИь, В клУЧЕНИь И сРАВНЕНИь. ОБЩИЕ ФУНкцИОНАльНы Е пРОстРАНстВА. III (пРОстРАНстВАB _p,q ^g(x) ИF _p,q ^g(x) , 1     

On the best approximation of functions on the sphere in the metric ofL p (Sn), 1<p<∞

L _p , 1<p<, S ⁿ R ⁿ⁺¹,n1, , , .     

Decrease orders of theL p -moduli of continuity (0<p≦∞)

k — , 0<p, _k(p) — k- L _p ¹ ; >0, - , (0, ), (t)0 (t0), (t) ,t ^–(t) . _k(p), , 0<p1 _{k(p) (k– 1)p+1}, 1<p — _k , .     

A method for summing Fourier integrals of functions from H p (E 2n + ), 0 < p < ∞

Suppose that H ^p (E _2n ⁺ ) is the Hardy space for the first octant $$E_{2n}^ + = \{ z \in \mathbb{C}^n :\operatorname{Im} z_j > 0, j = 1, \ldots ,n\} $$ and P _? ^l (f, x), l > 0, is the generalized Abel-Poisson means of a function f ? H ^p (E _2n ⁺ ). In this paper, we prove the inequalities $$C_1 (l,p)\widetilde\omega _l (\varepsilon ,f)_p \leqslant \left\| {f(x) - P_\varepsilon ^l (f,x)} \right\|_p \leqslant C_2 (l,p)\omega _l (\varepsilon ,f)_p ,$$ where $\widetilde\omega _l (\varepsilon ,f)_p $ and ω _l (?, f) _p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

On the <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript> limits of subsets of the real line

We point out that it is consistent with ZFC that 2 ^ω > ℵ₁ and every subset of ℝ is the ω ₁ limit of a sequence of G _δ sets in ℝ. We prove also that assuming cov ( ) > ℵ₁, not every set in ℝ is the ω ₁ limit of a sequence of measurable sets. This solves two problems of T. Natkaniec and J. Wesołowska.      

(<Emphasis Type="Italic">p, λ</Emphasis>)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK _λ ^p (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK _λ ^p (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK _λ ^p (A) we prove that ⊕ _i≥0 Ext _A ⁱ (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK _λ ^p (A) under certain conditions.     

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.     

НАИЛУЧШЕЕ ПРИБЛИЖЕ НИЕ УГЛОМ И ПРИБЛИЖЕНИЕ УГЛОМ ИЗ СИНГУЛЯРНЫХ ИНТЕГРАЛОВ ФУНКЦИЙf∈L p (R n ),2<P<∞

. .     

On solvability of f(p(x)) = q(f(x)) for given real functions p,q

We investigate the solvability of functional equations f(p(x)) =  q(f(x)) for given functions p and q which are partially or completely defined on the set of all real numbers. For these investigations, we use methods for constructions of homomorphisms of mono-unary algebras. We can present a simple characterisation of solvability of the above equation in the case that p, q are strictly increasing and continuous functions. It gives, on the one hand, a practical use for a class of functional equations. On the other hand, it is a contribution to questions on topological conjugacy of monotonous real functions.     

Toeplitz operators on hardy spaceH p (S) with 0<p≤1

LetB ⁿ be the unit ball inC ⁿ ,S is the boundary ofB ⁿ . We letL ^p (S) denote the usual Lebesgue spaces overS with respect to the normalized surface measure,H ^p (B ⁿ ) is its usua holomorphic subspace.H ^p (S) denotes the atomic Hardy spaces defined in [GL]. LetPL ² (S)H ²(B ⁿ ) denote the orthogonal projection. For eachfL (S), we useM _f L ^p (S)L ^p (S) to denote the multiplication operator, and we define the Toeplitz operatorT _f =PM _f . The paper gives a characterization theorem onf such that the Toeplitz operatorsT _f and are bounded fromH ^p (S)H ^p (B ⁿ ) with 0<p1. Also several equivalent conditions are given.     

On the measurability of the conic sections’ family in the projective space<Emphasis FontCategory="NonProportional">P</Emphasis><Subscript>3</Subscript><Subscript>3</Subscript>

In this paper we prove that the non degenerate conic sections’ family in the projective spaceP ₃ is measurable.     

The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

On the heritability of the property <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">π</Emphasis></Subscript> by subgroups

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.     

Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

A characterization of the algebra <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>(Ω)

We study some properties of algebras of continuous functions on a locally compact space, these algebras being equipped with the topology defined by a family of multiplication operators (β-uniform algebras). We prove an analog of a theorem due to Sheinberg for β-uniform algebras [see Uspekhi Mat. Nauk, 32:5 (197) (1977), 203–204].     

On Some Properties of Functions in the Sobolev-Morrey-Type Spaces <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>p,a,ϰ,τ</Subscript><Superscript>l</Superscript></Stack>(<Emphasis Type="Italic">G</Emphasis>)

In the framework of embedding theory, we study some properties of the functions of the Sobolev-Morrey-type spaces and local smoothness of solutions to one class of quasielliptic equations.Original Russian Text Copyright © 2005 Nadzhafov A. M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 634–648, May–June, 2005.     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.