Weighted Best Local Approximation

In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the Lpspaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 are extensively used.     

Local rotundity structure of Cesàro-Orlicz sequence spaces

Some criteria for extreme points and strong U-points in Cesàro-Orlicz spaces are given. In consequence we find a Cesàro-Orlicz sequence space different from c₀ which has no extreme points. Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro-Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of SU-point, that is, the notion of the extreme point is not strong enough here.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

Nonlinear Approximation with Dictionaries. II. Inverse Estimates

In this paper, which is the sequel to [16], we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequality obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality.     

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener-Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener-Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.     

Hausdorff Dimension in Convex Bornological Spaces

For non-metrizable spaces the classical Hausdorff dimension is meaningless. We extend the notion of Hausdorff dimension to arbitrary locally convex linear topological spaces and thus to a large class of non-metrizable spaces. This involves a limiting procedure using the canonical bornological structure. In the case of normed spaces the new notion of Hausdorff dimension is equivalent to the classical notion.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

On Bivariate Smoothness Spaces Associated with Nonlinear Approximation

In recent years there have been various attempts at the representations of {\mbox multivariate} signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves of singularities like the Mumford–Shah functional and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further simplified remains open.     

Two subcategories of apartness spaces

We introduce the notion of a topological quasi-apartness space and the notion of a uniform quasi-apartness space, and construct an adjunction between the category of topological quasi-apartness spaces and the category of neighbourhood spaces, and an adjunction between the category of uniform spaces and the category of uniform quasi-apartness spaces.     

Korovkin-type approximation in spaces of vector-valued and set-valued functions

We consider some Korovkin-type approximation results for sequences of linear continuous operators in spaces of vector-valued and set-valued continuous functions without assuming the existence of the limit operator. Even in spaces of real continuous functions, where similar results have already been established, we replace the positivity assumption with a weaker condition. We also give some quantitative estimate of the convergence and some applications where previous results cannot be applied.     

The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

REPELLERS FOR MULTIFUNCTIONS OF SEMI-BORNOLOGICAL SPACES

In this article the notion of repeller for multifunctions from the viewpoints of semi-bornologicul spaces is considered. The concept of lower semi-continuous multifunctions is extended by the use of semi-bornological spaces. Semi-bornological vector spaces are studied. The notion of conjugacy for semi-bornological multifunctions is considered. The persistence of repeller under conjugate relation is proved.     

Generalized Cauchy Spaces

In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.     

Properties of Θs-US spaces

Wilansky [6] introduced the notion of US spaces. It is the purpose of this paper to introduce and investigate the notion of Θs-US spaces.     

逼近性质和广义快速衰减

将快速衰减性质推广到可以一致嵌入到Hilbert空间的离散空间, 证明了一些逼近性质和应用.     

On mappings which approximately preserve angles

In this paper, we study mappings, which approximately preserve angles between inner product spaces. We also introduce a notion of angle in normed spaces. The notion of angle, considered in this part, relates to the well-known Birkhoff–James orthogonality. Based on it, we express a characterization for approximate Birkhoff–James orthogonality, introduced in the literature, through this notion of angle. Then we return to the issue of mappings which approximately preserve angle stating some results in normed spaces.     

Recurrent Linear Operators

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.     

P-Regular Convergence Spaces

A generalized notion of regularity is introduced which enables one to study locally compact spaces, sequential spaces, ω-regular spaces, and other diverse types of spaces as special wises of p-regular spaces.     

Violator spaces vs closure spaces

The main goal of the paper is to make connections between two well-known, but, up to now, independently developed theories: the theory of violator spaces and the theory of closure spaces. Violator spaces were introduced by Matoušek et al. in 2008 as generalization of linear programming problems. The notion of closure arises in many disciplines, including topology, algebra, convexity analysis, logic etc. In this work, we investigate interrelations between violator spaces and closure spaces. We show that a violator mapping may be defined by a weak version of a closure operator. Interrelations between violator spaces and closure spaces give new insights on a number of well known findings. For example, we prove that violator spaces with a unique basis satisfy both the anti-exchange and the Krein–Milman properties.Finally, based on subsequent relaxations of the closure operator notion we introduce convex spaces as a generalization of violator spaces.     

Uniformity Structures on L-Fuzzy Spaces

We extend the notion of a uniform space in a natural way by defining a uniform spaces in L-fuzzy spaces.Although these spaces seem quite similar to ordinary case,we show that the category of this uniform spaces is a good extension of the category of ordinary uniform spaces and the category of L-uniform spaces.Moreover,we introduce the concept of uniform topological spaces in the framework of uniform spaces in L-fuzzy spaces.Furthermore,the relation between proximity and uniform spaces in L-fuzzy spaces will...