Weak-star Topology Research Articles

Banach spaces not containing l1

We show that ifE is a complex Banach space which contains no subspace isomorphic tol1, then each infinite dimensional subspace ofE′ contains a normalized sequence which converges to zero for the weak star topology.

Continuity of the measure of maximal entropy for unimodal maps on the interval

LetT: [0, 1]→[0, 1] be a unimodal map with positive topological entropy. ThenT has a unique measure μ(T) of maximal entropy. It is proved that the mapT↦μ(T) is continuous with respect to the weak star-topology.

The structure of uniformly Gâteaux smooth Banach spaces

It is shown that a Banach spaceX admits an equivalent uniformly Gâteaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact.

Interpolation Methods for the Evaluation of 2π-Periodic Finite Baire Measure

We discuss the definition and effectiveness of a Pade-type approximation to 2π-periodic finite Baire measures on [-π,π]. In the first two sections we recall the definitions and basic properties of the Padre-type approximants to harmonic functions in the unit disk and to Lp-functions on the unit circle. Section 3 deals with the extension of these definitions and properties to a finite 2π-periodic Baire measure. Finally, section 4 is devoted to a study of the convergence of a sequence of such approximants, in the weak star topology of measures.

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC 1-mapX→ℝ for which {c∈X: T′c=0} is finite. Set $$T\left( T \right) = \cap _{j = 1}^\infty \overline {T^{ - j} X} $$ . Fix ann ∈ ℕ. For e>0, theC 1-map $$\tilde T:X \to \mathbb{R}$$ is called an e-perturbation ofT if $$\tilde T$$ is a piecewise monotonic map with at mostn intervals of monotonicity and $$\tilde T$$ is e-close toT in theC 1-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation $$\tilde T$$ ofT has a unique measure $$\tilde \mu $$ of maximal entropy, and the map $$\tilde T \mapsto \tilde \mu $$ is continuous atT in the weak star-topology.

Optimal control applications in transport theory

The critical neutron flux flattening problem governed by the transport equation is studied in a nonuniform slab with vacuum boundary conditions. Existence and uniqueness theorem of the optimal solution is shown in continuous function space. And it is shown that the continuous-space discrete-ordinates approximation converges to the exact optimal solution while the corresponding controllable function converges in the sense of weak star topology.

Completely contractive representations for some doubly generated antisymmetric operator algebras

Contractive weak star continuous representations of the Fourier binest algebra A \mathcal A (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of A \mathcal A by semicrossed product algebras A ( D ) × Z + A(\mathbb D)\times \mathbb Z_+ and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy–Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.

Compactness properties of the integration map for Volterra measures in non-normable spaces

We make a detailed study of the compactness properties of the integration map associated with the classical Volterra (vector valued) measure inX=L p([0, 1]), forp in [1, ∞], and inX=C([0, 1]), whereX is equipped with its relevant weak or weak-star topology. The spaceX (a Frechet space) of all complex valued sequences, equipped with the pointwise convergence topology, is also considered.

Generalized Szegö Theory in Frequency Analysis

A sequence of positive Borel measures with infinite support on the unit circle is given, converging in the weak star topology to a measure with support consisting of a finite numbern0of mass points. The sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied, with the focus being on the asymptotic behavior of the zeros. The main result is twofold: For a fixedn>n0,n0of the zeros (“interesting zeros”) tend to the mass points of the limiting measure, while the remainingn−n0zeros (“uninteresting zeros”) may not converge at all. However, by considering subsequences, limits may be obtained and the remaining zeros (and their limits) will be located in a closed disk |z|≤R<1, provided that a certain boundedness condition for the norm of the monic orthogonal functions is satisfied. As an application it is proved that this theory may be applied in frequency analysis, where the interesting zeros tend to the frequency points while the uninteresting zeros are bounded away from the unit circle.

Optimal control problems in spaces of functions of bounded variation

We study optimal control problems governed by differential systems involving measures as controls and where the state vector is a function of bounded variation. Such problems are defined as extensions to $BV$-spaces of classical problems which do not admit solutions in the class of absolutely continuous functions. We give a definition of generalized solutions for differential systems with measures, for which we prove a stability result for the weak-star topology of measures. We next prove existence of $BV$-solutions for control problems. A relaxation theorem is given: the classical control problem defined for $AC$-functions and the control problem extended to $BV$-spaces have the same value and the solutions of extended problem are cluster points of minimizing sequences for the initial problem. We finally characterize $BV$-solutions of control problems by means of Lipschitz solutions of an auxiliary control problem.

Fréchet Spaces of Holomorphic Functions without Copies of l 1

Let X be a Banach space. Let Hw*(X*) the Frechet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of Hw*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l1 when this space contains no copy of l1. We also show that if X is an Asplund space, then Hw*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.

Banach spaces of polynomials without copies of 𝑙¹

Let X be a Banach space. For a positive integer m, let P w ∗ ( m X ∗ ) {\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast }) denote the Banach space formed by all m-homogeneous polynomials defined on X ∗ {X^ \ast } whose restrictions to the closed unit ball B ( X ∗ ) B({X^ \ast }) of X ∗ {X^ \ast } are continuous for the weak-star topology. For each one of such polynomials, its norm will be the supremum of the absolute value in B ( X ∗ ) B({X^ \ast }) . In this paper the bidual of P w ∗ ( m X ∗ ) {\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast }) is constructed when this space does not contain a copy of l 1 {l^1} . It is also shown that, whenever X is an Asplund space, P w ∗ ( m X ∗ ) {\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast }) is also Asplund.

Lower Semicontinuity and Integral Representation of Functionals in BV([a, b]; Rm)

We obtain new lower semicontinuity results for an integral functional on the space BV([a, b]; Rm) endowed with its weak-star topology. We also prove, under some estimates on the integrand of the functional, that this functional is the lower semicontinuous regularization (in BV([a, b]; Rm)-weak*) of a classical functional defined on W1,1(a, b; R(m)).

The dual of a Gâteaux smooth Banach space is weak star fragmentable

The techniques of Preiss-Phelps-Namioka is used to prove that if a Banach space E E admits a Gâteaux smooth norm then its dual E ∗ {E^*} , endowed the weak star topology, is fragmentable.

Oscillations non linéaires des systèmes hyperboliques: méthodes et résultats qualitatifs

One studies the Cauchy problem for hyperbolic systems of first order conservation laws ∂tu + ∂xf(u) = 0, with an oscillating sequence of initial data. The oscillations have an O(1) amplitude; one may think to uε(x,0)=a(x;xε), where a(x,.) is periodic.It turns out that one part of the initial oscillations are killed because of genuine non-linearity. But the most of the physically relevant systems are endowed with at least one linearly degenerate characteristic field, which allows oscillations of a part of the field u.Our goal is to describe these oscillations and their propagation. There are several steps:–to determine the possible oscillations, x and t being fixed;–to parametrize them via a set of new variables (v(x, t), w(x, t; y)); here y plays the role of a “fast variable”;–to find an integro-differential system describing the evolution of (v, w);–to find appropriate initial data.Of particular interest is the computation of the group speeds, which are nothing but the characteristic values of the relaxed system. Formulas are given, especially for the full gas dynamics, which are all but trivial.The description of oscillations has been asked by Ronald DiPerna in 1985, who introduced the similar concept of “measure-valued solutions”. An equivalent problem is to determine whether the set of solutions is closed or not in the weak-star topology of L∞. Thus this work is an attempt to understand well-posedness (or illposedness, the both occur depending on the system) in L∞, one of the two classes where a ferv existence results are known.This study has also implications for numerical analysis. It suggests multi-scale methods which would give accurate values of speed when mild oscillations occur.The paper is organized into three parts. Section II-V are devoted to a rigourous analysis with applications. Sections VI-VIII concerns speculative (but efficient) ideas; this heuristic procedure gives the first description of the oscillations in gas dynamics and elastic string. The end of the article is concerned with a conjecture I made in 1983 about the compensated compactness method.

Closed ideals in the bidisc algebra

Let D be the open unit disc in C, T = 0 D the unit circle, and let D " = D X ... • be the n-dimensional unit polydisc. Let H ~ (D") be the algebra of bounded analytic functions on D", endowed with the uniform norm on D". The polydisc algebra is the space A(D")=C(D")nH~'(D"), also given the uniform norm on D". The spaces A(D) and A(D 2) are known as the disc and bidisc algebras, respectively. Let us introduce a weak-star topology on H = (D"). The space L ~* (T") is the dual space of LI(T"), so it has a weak-star topology. One can think of H*~(D ") as a subspace of L = (T") via radial limits, and as such it is weak-star closed. We define the weak-star topology on H = ( D ") by saying that a set U c H ~ ( D ") is open i f there is a weak-star open set V c L ~ ( T ") with U= VnH~ (D"). For a collection ~" o f functions in A(D"), associate the zero set

A measure which is singular and uniformly locally uniform

An example is given of a singular measure on [ 0 , 1 ] [0,1] which is locally nearly uniform in the weak star topology. If this measure is used as a prior to estimate an unknown probability in coin tossing, the posterior is asymptotically normal.

Unimodular functions and uniform boundedness

In this paper we study the role that unimodular functions play in deciding the uniform boundedness of sets of continuous linear functionals on various function spaces. For instance, inner functions are a UBD-set in H8 with the weak-star topology.

Banach spacesX withX** separable

LetX be a Banach space and letZ be a closed subspace ofX** which containsX. It is proved in this paper that, in the caseX** separable, there exists a closed subspaceY ofX such thatX+\(\bar Y\)=Z,\(\bar Y\) the closure ofY inX** for the weak-star topology.

An extension of Skorohod’s almost sure representation theorem

Skorohod discovered that if a sequence Q n {Q_n} of countably additive probabilities on a Polish space converges in the weak star topology, then, on a standard probability space, there are Q n {Q_n} -distributed f n {f_n} which converge almost surely. This note strengthens Skorohod’s result by associating, with each probability Q Q on a Polish space, a random variable f Q {f_Q} on a fixed standard probability space so that for each Q Q , (a) f Q {f_Q} has distribution Q Q and (b) with probability 1, f P {f_P} is continuous at P = Q P = Q .