The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Norm convergence of normalized iterates and the growth of Kœnigs maps

Let ? be an analytic function defined on the unit diskD, with ?(D)?D, ?(0)=0, and ?′(0)=λ≠0. Then by a classical result of G. K?nigs, the sequence of normalized iterates Φ _n /λ ⁿ converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH _p , the sequence Φ _n /λ ⁿ converges to σ in the norm ofH _p . We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ? is univalent. When ? is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyH _p . It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceL _a ^p .     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

Existence and non-existence of positive solutions of the scalar field system in R n,I

Letn > 3 andΩ be either the entire spaceR ⁿ or a Euclidean ball in R ⁿ . Consider the following boundary value problem (I) $$\{ _{\Delta v - u + u^q = 0,}^{\Delta u - v + v^p = 0,} u,v > 0, x \in \Omega $$ with homogeneous Dirichlet boundary data (replaced byu, v → 0 as ¦x¦ → ∞ when Ω=R ⁿ ), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if $$\frac{1}{{p + 1}} + \frac{1}{{q + 1}} > \frac{{n - 2}}{n}$$ . The existence on a ball and onR ⁿ are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence onR ⁿ .     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

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].     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space

We consider groups Γ generated by inversions in a pair of asymptotic complex hyperplanes in complex hyperbolic spaceH _? ⁿ . We show that there exists a Γ-invariant real hypersurfaceF ?H _? ⁿ such that the Dirichlet fundamental polyhedron for Γ centered at z₀ has two sides (resp. infinitely many sides) if and only ifz ₀ ∈F (resp.z ₀ ?F). The Dirichlet regions are determined explicitly in terms of coordinates on Γ-invariant horospheres and the geometry ofH _? ⁿ is developed in terms of these horospherical coordinates.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Spherical Means Associated with Non-isotropic Dilation Groups

Let {δ_t}_t>0 be a non-isotropic dilation group on R ⁿ . Let τ: R ⁿ → [0,∞) be a continuous function that vanishes only at the origin and satisfies τ(δ _t x) = tτ(x), t > 0, x ∈ R ⁿ . In this paper we obtain two-sided inequalities for spherical means of the form $\int_{S^{n-1}}\tau(r_1\omega_1,\cdots,r_n\omega_n)^{-\alpha}d\sigma (\omega),$ where α is a positive constant, and r₁,…, r_n are positive parameters.     

Approximation by spherical waves inL p -spaces

The paper is devoted to the following problem. Consider the set of all radial functions with centers at the points of a closed surface inR ⁿ . Are such functions complete in the spaceL ^q (R ⁿ )? It is shown that the answer is positive if and only ifq is not less than 2n/(n + 1). A similar question is also answered for Riemannian symmetric spaces of rank 1. Relations of this problem with the wave and heat equations are also discussed.     

Approximation of measures by Markov processes and homogeneous affine iterated function systems

It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system. We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR ⁿ can be approximated by invariant measures for finite iterated function systems of this class if \(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R ⁿ →R ⁿ . Moments and cumulants are also discussed.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.     

Approximation from shift-invariant spaces

We give sufficient conditions on a single function ? so that the principal shift-invariant space generated by ? provides a prescribed order of approximation inL _p (R ^d ), 1<p<∞, and inH _p (R ^d ), 0<p≤1. In particular, our conditions are given in terms of $\hat \varphi$ and are satisfied even when ? does not decay quickly at infinity.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

On Hardy-Bessel potential spaces over the ring of integers in a local field

Штрихарц [3] дал характе ристику пространствL ^p (R ⁿ ) бесселевых потенциа лов порядка α функций из п ространстваL ^p (R ⁿ ) с пом ощьюL ^p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF _α ^p (P ⁰) (0<p<1, α>1/р?1) в терминах функц ионаловS _r ^α f(x) (1τ≤2), которые в случаеR ⁿ } соответствуютD _α f (x), гдеР ⁰ - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов.