Relatively coarse sequential convergence

We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion of the group of rational numbers (equipped with the usual metric convergence) is complete; (ii) there are exactly exp exp such completions; (iii) the real line is the only one of them the convergence of which is Fréchet. Analogous results hold for the relatively coarse dense field precompletions of the subfield of all complex numbers both coordinates of which are rational numbers.     

On the convergence of constrained minima

We characterize the convergence of values and Lagrange multipliers for minimum problems of perturbed quadratic functionals in Banach spaces subject to a fixed linear operator constraint with finite-dimensional range. We obtain sufficient conditions for the convergence of the solutions. Examples show the different behaviour of the continuous dependence problem for the analogous unconstrained minimizations, where the gamma-type variational convergences are relevant. Such convergence conditions are extended here to the con strained problems.     

Sequential convergence via Galois correspondences

Topological sequential spaces are the fixed points of a Galois correspondence between collections of open sets and sequential convergence structures. The same procedure can be followed replacing open sets by other topological concepts, such as closure operators or (ultra)filter convergences. The fixed points of these other Galois correspondences are not topological spaces in general, but they can be embedded into the larger topological classes of pretopological, pseudotopological and convergence spaces. In this paper, we characterize the sequential convergences which are fixed points of these correspondences as well as their restrictions to topological spaces.     

G-Uniformities,LR-Proximities and Hypertopologies

We look closely at the relationships between hit-and-miss and proximal hit-and-miss -topologies, in the setting of proximity spaces. We provide equivalent conditions that force comparisons among proximal hit-and-miss -topologies determined by different proximities. We pay attention to these topologies when consists of the family of all closed balls of a proximity space, and we study their interplay with the Wijsman convergence expressed in proximity spaces. Finally we study the supremum of all Wijsman convergences and of all proximal ball topologies when X is at most regular, and the infimum of all Wijsman convergences when X is at least Tychonoff.     

Unbounded norm convergence in Banach lattices

A net \((x_\alpha )\) in a vector lattice X is unbounded order convergent to \(x \in X\) if \(|x_\alpha - x| \wedge u\) converges to 0 in order for all \(u\in X_+\). This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net \((x_\alpha )\) in a Banach lattice X is unbounded norm convergent to x if Open image in new window for all \(u\in X_+\). We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.     

Variational convergence over metric spaces

We introduce a natural definition of -convergence of maps, , in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the -convergence, we establish a theory of variational convergences. We prove that the Poincaré inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are -spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.

     

Pointwise convergence fails to be strict

It is known that the ring of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of . In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of which differs from .     

Compactness in Categories of S-Net Spaces

The $S$ -net spaces studied are convergence structures whose convergences are expressed by using generalized nets, the so called $S$ -nets, which are obtained from the usual nets by replacing the category of directed sets and cofinal maps with an arbitrary construct $S$ . We investigate compactness in categories of $S$ -net spaces defined by introducing continuous maps in a natural way and imposing some usual convergence axioms.     

On smoothness conditions and convergence rates in the CLT in Banach spaces

Summary In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n^–1/2) for sets which have regular boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRGResearch supported by the SFB 343 at Bielefeld     

Weak set convergence and its application to projection methods

There are many examples in Numerical Analysis where convergence of approximate solutions to a solution of the original problem can not be shown in the sense of a norm topology but in the sense of weak convergence ([6], [9], [10]).

Moreover, (global) solutions are often not unique such that a concept of set convergence instead of convergence in the usual sense is more convenient and reasonable ([1], [2]). This particularly holds if weakly formulated problems are under consideration.

When dealing with problems where both situations coincide, a concept of weak set convergence seems to be adequate. Such a concept is developed and will be applied to certain projections methods.     

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years.For an admissible ideal ∮N× N,the aim of the present paper is to introduce the concepts of ∮-convergence and ∮*-convergence for double sequences on probabilistic normed spaces(PN spaces for short).We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide.We also define ∮-Cauchy and ∮*-Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces.We establish example which shows that our method of convergence for double sequences on PN spaces is more general.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing-related properties of a complete Boolean algebra B with the properties of the convergences λ_s (the algebraic convergence) and λ_ls on B generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ_ls is a topological convergence iff forcing by B does not produce new reals and that λ_ls is weakly topological if B satisfies condition (?) (implied by the t-cc). On the other hand, if λ_ls is a weakly topological convergence, then B is a 2^h-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence λ_ls on the collapsing algebra \(B = {\text{ro}}{(^{ < \omega }}{\omega _2})\) is weakly topological” is independent of ZFC.     

On the convergence of rational approximations of semigroups on intermediate spaces

We generalize a result by Brenner and Thomée on the rate of convergence of rational approximation schemes for semigroups. Using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the Banach space and the domain of a certain power of the generator of the semigroup. The sharpness of the results is also discussed.

     

Thresholding in Learning Theory

In this paper we investigate the problem of learning an unknown bounded function. We will emphasize special cases where it is possible to provide very simple (in terms of computation) estimates enjoying, in addition, the property of being universal, i.e. their construction does not depend on the a priori knowledge of regularity conditions on the unknown object and still they have almost optimal properties for a whole group of functions spaces. These estimates are constructed using a thresholding technique, which has proven in the last decade in statistics to have very good properties for recovering signals with inhomogeneous smoothness but has not been extensively developed in learning theory. We will basically consider two particular situations. In the first case, we consider the RKHS situation, where we produce a new algorithm and investigate its performances in . The exponential rates of convergences are proved to be almost optimal, and the regularity assumptions are expressed in simple terms. The second case considers a more specified situation where the X_i's are one-dimensional and the estimator is a wavelet thresholding estimate. The results are comparable in this setting to those obtained in the RKHS situation, as concerned the critical value and the exponential rates. The advantage here is that we are able to state the results in the -norm and the regularity conditions are expressed in terms of standard Holder spaces.     

Hypertopologies on function spaces

The aim of this paper is to continue Naimpally’s seminal papers [16], [17], [18], i.e. we investigate topological properties of spaces which force the coincidence of convergences of functions associated with different hyperspace topologies. For example a metric spaceX is locally compact iff the topological convergence and the convergence induced by the Fell topology coincide onC(X,IR). Moreover, the proximal topology on the space of functions, not necessarily continuous, is studied in great detail.     

On some generalizations of Lorentz's almost convergence

We investigate an extension of the almost convergence of G.G. Lorentz, further weakening the notion of M-almost convergence we defined in [S. Mercourakis, G. Vassiliadis, An extension of Lorentz's almost convergence and applications in Banach spaces, Serdica Math. J. 32 (2006) 71–98] and requiring that the means of a bounded sequence restricted on a subset M of converge weakly in ℓ^∞(M). The case when M has density 1 is of special interest and in this case we derive a result in the direction of the Mean Ergodic Theorem (see Theorem 2).     

Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables (X _n ) we study functional weak convergence of partial maxima processes \(M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]\) in the space D[0, 1] with the Skorohod J ₁ topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J ₁ and M ₁ functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition \(\mathcal {A}(a_{n})\) with the time component.     

On the rate of convergence of certain random series in terms of norms of Orlicz spaces

There are obtained conditions for the convergence and estimates of the rate of convergence in terms of norms of Orlicz spaces of random series of the form where f_k(¯t) is an orthonormal system of eigenfunctions of a certain integral equation, k are random variables in an Orlicz space, ¯t Rⁿ.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 841–848, June, 1991.     

Flat convergence for integral currents in metric spaces

In compact local Lipschitz neighborhood retracts in weak convergence for integral currents is equivalent to convergence with respect to the flat distance. This comes as a consequence of the deformation theorem for currents in Euclidean space. Working in the setting of metric integral currents (the theory of which was developed by Ambrosio and Kirchheim) we prove that the equivalence of weak and flat convergence remains true in the more general context of metric spaces admitting local cone type inequalities. These include in particular all Banach spaces and all CAT(κ)-spaces. As an application we obtain the existence of a minimal element in a fixed homology class and show that the weak limit of a sequence of minimizers is itself a minimizer.