On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L ² (?π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (?∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L ² (?π, π) andE ² . When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities $$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$ hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ?^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(?∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence. Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].     

On subgroups of hypercentral type of finite groups

The main purpose of this paper is to analyze the influence on the structure of a finite group of some subgroups lying in the hypercenter. More precisely, we prove the following: Let \(\mathfrak{F}\) be a Baer-local formation. Given a group G and a normal subgroup E of G, let \(Z_\mathfrak{F} (G)\) contain a p-subgroup A of E which is maximal being abelian and of exponent dividing p ^k , where k is some natural number, k ≠ 1 if p = 2 and the Sylow 2-subgroups of E are non-abelian. Then E/O _p′ (E) ≤ \(Z_\mathfrak{F} \) (G/O _p′ (E)) (Theorem 1). Some well-known results turn out to be consequences of this theorem.     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

Asplund Operators and p-Integral Polynomials

We introduce the Banach ideals of p-integral and of p-nuclear polynomials for 1 ≤ p ≤ + ∞, extending to the polynomial setting the well known notions of p-integral and p-nuclear operators. For p = 1, we recover the Pietsch integral and nuclear polynomials, respectively. Given a Banach space E, let K be a compact Hausdorff space such that there is an embedding h : E → C(K). Let R _h be the polynomial from E into C(K) given by R _h (x) : = h(x) ^m for all ${x \epsilon E}$ . We prove that a polynomial is p-integral (1 ≤ p ≤ + ∞) if and only if it factors through a polynomial of the form R _h followed by the canonical inclusion of C(K) into L _p (K, μ) for some finite measure μ. We also prove that a polynomial P is p-integral if and only if we may write ${P = T \circ R_{h}}$ where T is a p-integral operator on a C(K) space. We show that P is ∞-integral if and only if it factors in the form ${P = T \circ R_{h}}$ where T is a weakly compact operator on a C(K) space. Analogous results are true if we replace C(K) by L _∞ (Ω, μ) for some finite measure space (Ω, Σ, μ). It is proved that a polynomial ${P \epsilon \mathcal{P}(^{m}E, F)}$ is p-integral if and only if its linearization is well defined and p-integral on ${\bigotimes ^{m}_{{\epsilon}_{s}}, s^{E}}$ . It is also shown that a p-integral polynomial may be extended to a p-integral polynomial on every larger space, and the extension has the same p-integral norm. We give a factorization theorem for p-nuclear polynomials. Finally, we prove that a polynomial P is p-nuclear if and only if it may be written in the form ${P = Q \circ A}$ where A is a compact operator and Q is a p-integral polynomial, if and only ${P = Q^{\prime} \circ H}$ with H an Asplund operator and Q′ a p-integral polynomial. This extends a result obtained by C. Cardassi in the linear case.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

Hausdorff dimension of Lebesgue sets for W α p classes on metric spaces

Let (X, μ, d) be a space of homogeneous type, where d and p are a metric and a measure, respectively, related to each other by the doubling condition with γ > 0. Let W _α ^p (X) be generalized Sobolev classes, let Cap_{α p} (where p > 1 and 0 < α ≤ 1) be the corresponding capacity, and let dim_H be the Hausdorff dimension. We show that the capacity Cap_{α p} is related to the Hausdorff dimension; we also prove that, for each function u ∈ W _α/p (X), p > 1, 0 < a < γ/p, there exists a set E ? X such that dim _H (E) ≤ γ - αp, the limit $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {u d\mu = u * (x)} $$ exists for each x ∈ X\E, and moreover $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {\left| {u - u * (x)} \right|^q d\mu = 0, \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{\gamma }.} $$ .     

Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

We consider a stochastic recursion X _n+1 = M _n+1 X _n  + Q _n+1, ( ${n\in \mathbb {N}}$ ), where (Q _n , M _n ) are i.i.d. random variables such that Q _n are translations, M _n are similarities of the Euclidean space ${\mathbb {R}^d}$ and ${X_n\in \mathbb {R}^d}$ . In the present paper we show that if the recursion has a unique stationary measure ν with unbounded support then the weak limit of properly dilated ν exists and defines a homogeneous tail measure Λ. The structure of Λ is studied and the supports of ν and Λ are compared. In particular, we obtain a product formula for Λ.     

Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p,where p is a prime number.As a consequence we prove if |G|=2δp,δ=0,1,2 and p prime,then Γ=Cay(G,S) is a connected normal 1/2 arc-transitive Cayley graph only if G=F4p,where S is an inverse closed generating subset of G which does not contain the identity element of G and F 4p is a group with presentation F4p = a,b|ap=b4=1,b-1ab=aλ,where λ2≡-1(mod p).     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph K _N contains a monochromatic copy of H. A famous result of Chváatal, Rödl, Szemerédi and Trotter states that there exists a constant c(Δ) such that r(H) ≤ c(Δ)n for every graph H with n vertices and maximum degree Δ. The important open question is to determine the constant c(Δ). The best results, both due to Graham, Rödl and Ruciński, state that there are positive constants c and c′ such that $2^{c'\Delta } \leqslant c(\Delta ) \leqslant ^{c\Delta \log ^2 \Delta }$ . We improve this upper bound, showing that there is a constant c for which c(Δ) ≤ 2 ^cΔlogΔ . The induced Ramsey number r _ind (H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd?s conjectured the existence of a constant c such that, for any graph H on n vertices, r _ind (H) ≤ 2 ^cnlogn . We move a step closer to proving this conjecture, showing that r _ind (H) ≤ 2 ^cnlogn . This improves upon an earlier result of Kohayakawa, Prömel and Rödl by a factor of logn in the exponent.     

Об условиях ограниченности полиномов на отрезке в случае их равномерной ограниченности на подмножестве зтого отрезка

It is proved that the inequality $$\rho (E) \le \frac{\theta }{{n^2 }} (\theta > 0, \rho (E) = \mathop {max}\limits_{x \in [ - 1,1]} \mathop {inf}\limits_{y \in E} |x - y|)$$ is a sufficient condition for the boundedness of a polynomial of degree ≤ n (n ≥17 max(?; 5)) on the whole segment [?1, 11, provided it is unformly bounded on the subset E of this segment. Furthermore, the above condition cannot be weakened (if all subsets E are taken into account).