Metric Of Uniform Convergence Research Articles

Methods of comparison of families of real functions in porosity terms

Abstract We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.

Large deviations for processes on half-line: Random walk and compound Poisson process

We establish, under the Cramer exponential moment condition in a neighbourhood of zero, the Extended Large Deviation Principle for the Random Walk and the Compound Poisson processes in the metric space $\V$ of functions of finite variation on $[0,\infty)$ with the modified Borovkov metric $\r(f,g)= \r_\B(\hat{f},\hat{g}) $, where $ \hat f(t)= f(t)/(1+t)$, $t\in \R$, and $\r_\B$ is the Borovkov metric. LDP in this space is more precise than that with the usual metric of uniform convergence on compacts.

Typical behaviour of integrable functions at infinity

We obtain the following extension of a theorem due to Lesigne. Let L1:=L1([0,∞)) and let C(1) be the (Polish) space of nonnegative continuous functions f on [0,∞) such that ∫[0,∞)f≤1, with the metric of uniform convergence on every compact subset of [0,∞). Denote c0+:={(bn)∈c0:bn>0 for all n∈N}. Then, for Y:=L1, the sets{(b,f)∈c0+×Y:lim supn→∞f(nx)bn=∞ for almost all x≥0},{f∈Y:lim supn→∞f(nx)bn=∞ for almost all x≥0}whereb∈c0+, are comeagre of type Gδ. If Y:=C(1), the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type Gδ.

Large deviations for processes on half-line

We consider a sequence of processes defined on half-line for all non negative t. We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with a new metric that is more sensitive to behaviour at infinity than the uniform metric. LDP is established for Random Walks, Diffusions, and CEV model of ruin, all defined on the half-line. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.

Approximate fixed points of nonexpansive mappings in unbounded sets

It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of R X to the space of real-valued continuous functions C ( X ) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C ( X ) . We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure H ¯ of H to C ( X ) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C ( X ) : here distance refers to the metric of uniform convergence in R X . We study the quantitative behavior of sequences in H approximating points in H ¯ . As a particular case we obtain the results known about angelicity for these C p ( X ) spaces obtained by Orihuela. We indeed prove our results for spaces C ( X , Z ) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.

More About Sierpiński-Zygmund Uniform Limits of Extendable Functions

Let $SZ$, $D$, $Ext$, and $\overline{Ext}$ denote respectively the spaces of Sierpiński-Zygmund functions, Darboux functions, extendable connectivity functions, and uniform limits of sequences of extendable connectivity functions, with the metric of uniform convergence on them. We show that the subspaces $SZ \cap D$ and $SZ \cap \overline{Ext}$ are each porous in the space $SZ$, but $SZ \cap \overline{Ext}$ is not porous in the space $\overline{Ext}$. We also show that every real function can be expressed as a sum of two Sierpiński-Zygmund functions one of which belongs to $\overline{Ext}$. Ciesielski and Natkaniec showed in 1997 that if $\R$ is not the union of less than $\mathfrak c$-many nowhere dense subsets, then there exist Sierpiński-Zygmund bijections $f,g:\mathbb{R}\to\mathbb{R}$ such that $f^{-1} \notin SZ$ and $g^{-1}\in SZ$, but here we can additionally have $f$ and $g$ belonging to $\overline{Ext}$

On topological properties of the spaces of Darboux Baire 1 functions and bounded derivatives

We investigate the topological structure of the space ${\mathcal D}{\mathcal B}_{1}$ of bounded Darboux Baire 1 functions on $[0,1]$ with the metric of uniform convergence and with the $p^{*}$-topology. We also investigate some properties of the set ${\

Properties of the Space DB1∗∗ with the Metric of Uniform Convergence

Properties of the Space DB1∗∗ with the Metric of Uniform Convergence

Negligible subsets in the space of homeomorphisms

We show that for various compact metric spaces X, the space of homeomorphisms H( X) is homeomorphic to H( X)⧹ K, where K=⋃ i>0 K i ⊂ H( X) where either (1) each K i is closed and equi-uniformly continuous or (2) each K i is complete in the metric of uniform convergence.

ON SMALL SUBSETS OF THE SPACE OF DARBOUX FUNCTIONS

We prove that the set F of all bounded functionally connected functions is boundary in the space of all bounded Darboux functions (with the metric of uniform convergence). Next we prove that the set of bounded upper (lower)semi-continuous Darboux functions and the set of all bounded quasi-continuous functionally connected functions is porous at each point of the space F . Since 1875, when Darboux functions were defined, many papers have appeared about their properties. Proofs of many properties of real Darboux functions of real variables are very important, because the family of Darboux functions includes many important classes of functions; for example, continuous functions and functionally connected functions. (See the next page for the definition.) Mutual inclusions among several families of Darboux-like functions inspired questions concerning the size of particular sets in spaces of functions. One of the questions is how strong the inclusions are. Similar issues have already been considered in a great number of papers such as [4], [5] and [6]. In [3] Jȩdrzejewski noticed that each continuous function is functionally connected and each functionally connected function is a Darboux function. It is not difficult to show that there exists a discontinuous, functionally connected function and there exists a Darboux function which is not functionally connected. Moreover, it turns out that in the space of bounded Darboux functions (with the metric of uniform convergence) the set of bounded functionally connected functions is boundary (its complement is a dense set) and in the space of bounded functionally connected functions the set of bounded upper (lower)

On some properties of the spaces of almost continuous functions

In this paper we prove that, in the space 𝒜 of almost continuous functions (with the metric of uniform convergence), the set of functions of the first class of Baire is superporous at each point of this space

On the porosity of the set of 𝜔-nonexpansive mappings without fixed points

Let C C be a nonempty closed convex bounded subset of a Banach space E E . Let M \mathcal {M} denote the family of all multivalued mappings from C C into E E which are nonempty weakly compact convex valued, ω \omega -nonexpansive and weakly-weakly u.s.c., endowed with the metric of uniform convergence. Let M 0 {\mathcal {M}_0} be the set of all F ∈ M F \in \mathcal {M} for which the fixed point problem is well posed. It is proved that the set M ∖ M 0 \mathcal {M}\backslash {\mathcal {M}_0} is σ \sigma -porous (in particular meager). A similar result is given for weak properness.

ON THE QUESTION OF UNIVERSAL INTEGRABILITY OF BOUNDED FUNCTIONS

Various procedures are considered for integration of bounded real-valued functions with respect to finitely additive measures on a semi-algebra of sets. A criterion is established for the indefinite Darboux integrals of a bounded function to coincide on the family of all positive finitely additive measures: a bounded function is universally integrable if and only if it belongs to the closure, in the metric of uniform convergence, of the linear span of the family of all characteristic functions of sets in the semi-algebra. A representation of the indefinite Darboux integral is obtained for such bounded functions. For arbitrary bounded functions a construction is proposed for a multivalued indefinite integral with respect to a positive finitely additive measure, and some of the properties of the construction are established. In particular, the multivalued integral of an arbitrary bounded function with respect to a positive countably additive measure consists only of countably additive measures of bounded variation, while the multivalued integral with respect to a purely finitely additive positive measure consists only of purely finitely additive measures. The dependence of the multivalued integral on the bounded function is continuous in the sense of the natural metric for the space of nonempty order intervals in the family of finitely additive measures of bounded variation.Bibliography: 7 titles.

Simultaneous Automorphisms in Spaces of Analytic Functions

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .