The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x_n) → x in X and (x_n*) → 0 in X* with ||x_n|| = ||x||= 1 we have (x_n*(x_n)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S^th are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.     

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L⁺(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L⁺(D) is well-ordered by inclusion and the α-th closed ideal ??_α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then ??_α. An element A in L⁺(D) belongs to the minimal closed ideal ??₀ if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.     

Energy concentration for the Landau–Lifshitz equation

For the Landau–Lifshitz equation on a domain with three space dimensions, we consider energy concentration phenomena arising in the context of weakly convergent sequences of solutions. The concentration measure can be interpreted as a family of generalized curves. We establish a connection to a geometric flow.     

Weak density of smooth maps for the Dirichlet energy between manifolds

We prove that given a simply connected compact manifold M and a closed manifold N, any map in the Sobolev space W ^1,2(M,N) can be approximated weakly by smooth maps between M and N. Submitted: September 2002, Final version: November 2002.     

Weak semi-continuity of the duality product in Sobolev spaces

Given a weakly convergent sequence of positive functions in , we prove the equivalence between its convergence in the sense of obstacles and the lower semi-continuity of the term by term duality product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of . This result implicitly gives new characterizations for both the convergence in the sense of obstacles of a weakly convergent sequence of positive functions and for the weak l.s.c. of the duality product.     

Weakly convergent sequences of functions and orthogonal polynomials

Measures and sequences of functions on locally compact spaces are studied, and a condition is given that, for a sequence of functions that is weakly convergent in L¹, ensures the strong convergence of a related sequence of functions. This result, together with a new integral formula for the reflection coefficients Φ_n(0) for the monic orthogonal polynomial Φ_n associated with a measure on the unit circle, is used to investigate convergence properties of orthogonal polynomials.     

Tauberian conditions for w-almost convergent double sequences

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L^p(T²), where 1 < p < ∞.      

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

On some Banach space properties sufficient for weak normal structure and their permanence properties

We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.

     

Topological Spaces with Skorokhod Representation Property

We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1171–1186, September, 2005.     

Ultra-parabolic H-measures and compensated compactness

We present a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints. The main tool is the localization properties for ultra-parabolic H-measures corresponding to weakly convergent sequences.     

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

Central limit theorems for the ergodic adding machine

In this paper we find a second class of sequences of random numbers (x _n ) _n=1 ^∞ (the orbit of the ergodic adding machine) such that the corresponding sequences of zeros and ones 1_[0,y](x _n) (n=1,2,...,N) satisfy Central Limit Theorems with extremely small standard deviationσ _N=O(√logN), instead ofO(√N), asN → ∞. Dedicated to Professor Benjamin Weiss on the occasion of his 60^th birthday.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Weak statistical convergence and weak filter convergence for unbounded sequences

For every weakly statistically convergent sequence (xn) with increasing norms in a Hilbert space we prove that . This estimate is sharp. We study analogous problem for some other types of weak filter convergence, in particular for the Erdös-Ulam filters, analytical P-filters and Fσ filters. We present also a refinement of the recent Aron-Garcia-Maestre result on weakly dense sequences that tend to infinity in norm.     

Energy quantization for triholomorphic maps

Let be weakly convergent stationary triholomorphic maps from a hyperkähler manifold M to another hyperkähler manifold N. We establish an energy quantization for the density function of the defect measure on the concentration set.Received: 10 July 2002, Accepted: 30 September 2002, Published online: 17 December 2002     

Young measures, weak and strong convergence and the Visintin-Balder Theorem

This paper is concerned with sequences inL ¹ which converge weakly. Young's measures theory permits us to give sufficient conditions insuring the strong convergence and to understand the behaviour of the sequences which do not converge strongly.     

On weak convergence of functionals of stochastic processes with continuously differentiable paths

We obtain a criterion for weak convergence of a sequence of stochastic processes _n(t), t [0, 1],n N, _n(t) R ^m in the spaceC _m ^k [0, 1] of continuously differentiable functions. We consider several examples of weakly convergent sequences of stochastic processes inC _m ^k [0, 1] and several integer functionals defined on these random variables.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 85–90, 1987.     

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x_n) in X, there exists a subsequence (x_k(n)) so that (Tx_k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if is collectively weakly compact, then M^* is weakly equicompact iff M^** x^**={T^** x^** : T ∈ M} is relatively compact in Y for every x^** ∈X^**; 2) weakly equicompact sets are precompact in for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005     

A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation

In the case of the Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with a small parameter ɛ multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost ɛ-uniformly (that is, the convergence rate of this scheme weakly depends on ɛ) is constructed. When ɛ is not very small, this scheme converges with an order of accuracy close to two. For the construction of the scheme, we use the classical monotone (of the first order of accuracy) approximations of the differential equation on a priori adapted locally uniform grids that are uniform in the domains where the solution is improved. The boundaries of such domains are determined using a majorant of the singular component of the grid solution. The accuracy of the scheme is improved using the Richardson technique based on two embedded grids. The resulting scheme converges at the rate of O((ɛ⁻¹ N ^−K ln² N)² + N ⁻²ln⁴ N + N ₀⁻²) as N, N ₀ → ∞, where N and N ₀ determine the number of points in the meshes in x and in t, respectively, and K is a prescribed number of iteration steps used to improve the grid solution. Outside the σ-neighborhood of the lateral boundary near which the boundary layer arises, the scheme converges with the second order in t and with the second order up to a logarithmic factor in x; here, σ = O(N ^{−(K − 1)}ln² N). The almost ɛ-uniformly convergent finite difference scheme converges with the defect of ɛ-uniform convergence ν, namely, under the condition N ⁻¹ ≪ ɛ^ν, where ν determining the required number of iteration steps K (K = K(ν)) can be chosen sufficiently small in the interval (0, 1]. When ɛ⁻¹ = O(N ^{K − 1}), the scheme converges at the rate of O(N ⁻²ln⁴ N + N ₀⁻²).