Optimal decompositions for theK-functional for a couple of Banach lattices

Letf=g _t+h _t be the optimal decomposition for calculating the exact value of theK-functionalK(t, f; ) of an elementf with respect to a couple =(X ₀ ,X ₁) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spacesX ₀ andX ₁ is eitherL ^∞ orL ¹. Many examples are given of couples of lattices for which |g _t| increases monotonically a.e. with respect tot. It is shown that this property implies a sharpened estimate from above for the Brudnyi-KrugljakK-divisibility constant γ( ) for the couple. But it is also shown that certain couples do not have this property. These also provide examples of couples of lattices for which γ( ). Research supported by the Technion V. P. R. Fund.     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.     

Irreducibility of the punctual quotient scheme of a surface

It is shown that the punctual quotient schemeQ _l ^r parametrizing all zero-dimensional quotients of lengthl and supported at some fixed point O∈A ² in the plane is irreducible.     

Vector-valued Hardy inequalities andB-convexity

Inequalities of the form for allf∈H ¹, where {m _k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that satisfies the previous inequality for vector valued functions inH ¹ (X), defined as the space ofX-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {m _k }={2^k} but not for {m _k }={k ^a } for any α ∈ N. The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0291.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

Projections in the space<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

LetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH ₁(M,Z)=H ₁(R,Z). Let be a covering. We study the algebraH ^∞(U) of bounded holomorphic functions defined in certain subdomains . Our main result is a Forelli type theorem on projections inH ^∞(D). Research supported in part by NSERC.     

Tracking and invariance problems for some classes of discrete delay systems

Consider the system with perturbation g _k ∈ ℝ ⁿ and output z _k = Cx _k . Here, A _k ,A _k ^(s) ∈ ℝ ^{n × n} , B _k ⁽¹⁾ ∈ ℝ ^{n × p} , B _k ⁽²⁾ ∈ ℝ ^{n × m} , C ∈ ℝ ^{p × n} . We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u _k ⁽¹⁾ and u _k ⁽²⁾ for which the following properties are satisfied:

On the spectral representation of the elementary operator and spectral mapping theorems

LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X) ⁿ ,B∈B(Y) ⁿ , the elementary operator acting onB(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS _p (L _A ,R _B )=σ(A)×σ(B) and .     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Topological ergodic decompositions and applications to products of powers of a minimal transformation

For a minimal distal flow (X, T) and a positive integern, let be the largest distal factor of ordern. The existence of a denseG _δ subset ω ofX is shown, such that forx ∈ ω the orbit closure of (x,x,...,x) ∈ X ⁿ⁺¹ under τ =T ×T ² ... ×T ⁿ⁺¹ is π-saturated. In fact, an analogous statement for a general minimal flow is proved in terms of its PI-tower. On the way we get some topological “ergodic” decomposition theorems.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

Integral inequalities for self-reciprocal polynomials

Let n ≥ 1 be an integer and let P _n be the class of polynomials P of degree at most n satisfying z ⁿ P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P _n :

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that

Régularisation spectrale et propriétés métriques des moyennes mobiles

We develop a spectral regularization technique for moving averages , where ϕ is a nondecreasing map andU: H→H is a contraction of a Hilbert space (H, ‖·‖). We obtain a spectral regularization inequality which allows one to evaluate efficiently the increments ‖B _m ^U , ϕ (f)−B _n ^U , ϕ (f)‖,f∈H, by means of where is a properly regularized version of the spectral measure off with respect toU. We apply this inequality to an investigation of metric properties of the sets of moving averages {B _n ^{U, ϕ} (f), n ∈N} with fixedf∈H andN⊂ N. In particular, we obtain estimates of the associated covering numbers as well as of the related Littlewood-Paley-type square functions. This work extends our previous results concerning the case of classical averages (ϕ(n)=0). Since it is well-known that the structure of general moving averages is more complicated, there is no surprise that the general results we obtain are sometimes less complete than their classical counterparts and need suitable moment assumptions on the spectral measure (depending on the growth of the shift function ϕ). Nevertheless, when applied to the classical situation, our estimates still yield optimal bounds.

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Avec pour le premier author, le soutien de la fondation russe pour la recherche fondamentale, subvention 99-01-00112 et INTAS subvention 99-01317.     

Théorèmes d’approximation pour les sections différentiables

Resume Le principail résultat de ce travail est un Théorème d’approximation des sections différentiables d’un fibré linéaireF sur un espace analytique réel cohérentX. (I) On définit le concept de sections différentiables d’un tel fibré puis le faisceau des germes de sections différentiablesC ^∞ (F) qui s’identifie àF ^∞=F⊗ _Ox C _X ^∞ :F=dual du fibré linéaireF,C _X ^∞ =faisceau des germes de fonctions différentiables surX. (II)X est supposé en plus paracompact et localement compact. On construit surF ^∞(X) une ?topologie de Whitney? pour laquelleF(X) est dense dansF ^∞(X). La démonstration utilise les complexifiés et et les voisinages de Stein deX dans .

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Sunto Il risultato principale di questo lavoro è un teorema di approssimazione delle sezioni differenziabili di un fibrato lineareF su uno spazio analitico reale coerenteX. (I) Si definisce il concetto di sezione differenziabile di un tale fibrato poi il fascio dei germi delle sezioni differenziabiliC ^∞ (F) che si identifica aF ^∞=F⊗ _Ox C _X ^∞ :F=duale del fibrato lineareF,C _X ^∞ dei germi delle funzioni differenziabili suX. (II)X è supposto inoltre paracompatto e localmente compatto. Si costruisce suF ^∞ (X) una ?topologia di Whitney? per la qualeF(X è denso inF ^∞(X). La dimostrazione utilizza i complessificati e e gli intomi di Stein diX in .

     

Rationally convex sets on the unit sphere in ℂ2

Let X be a rationally convex compact subset of the unit sphere S in ?², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the $\bar{\delta}_{b}Let X be a rationally convex compact subset of the unit sphere S in ℂ², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3]. We define a real-valued function ε _X on the open unit ball intB, with ε _X (z,w) tending to 0 as (z,w) tends to X. We give a growth condition on ε _X (z,w) as (z,w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in intB. For each compact set Y in ℂ², we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1].     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.