Measure theory of statistical convergence

The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)     

The Banach Principle for ideal convergence in the classical and noncommutative context

Versions of the Banach Principle for different types of convergence ‘with respect to an ideal’ are established both in the commutative and noncommutative (von Neumann algebraic) context.     

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.     

Weak and point-wise convergence in for filter convergence

We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study.     

A remark about the Orlicz-Pettis theorem and the statistical convergence

In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.     

B值随机元阵列的矩完全收敛性

在随机元阵列随机有界于某非负随机变量的条件下,得到了B值行独立的随机元阵列的矩完全收敛性的一些充分条件.同时研究了p型Banach空间中行独立的随机元阵列的矩完全收敛性.     

A certain class of weighted statistical convergence and associated Korovkin‐type approximation theorems involving trigonometric functions

The subject of statistical convergence has attracted a remarkably large number of researchers due mainly to the fact that it is more general than the well‐established theory of the ordinary (classical) convergence. In the year 2013, Edely et al 17 introduced and studied the notion of weighted statistical convergence. In our present investigation, we make use of the (presumably new) notion of the deferred weighted statistical convergence to present Korovkin‐type approximation theorems associated with the periodic functions , and defined on a Banach space . In particular, we apply our concept of the deferred weighted statistical convergence with a view to proving a Korovkin‐type approximation theorem for periodic functions and also to demonstrate that our result is a nontrivial extension of several known Korovkin‐type approximation theorems which were given in earlier works. Moreover, we establish another result for the rate of the deferred weighted statistical convergence for the same set of functions. Finally, we consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.     

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.     

Weak convergence conditions for Inexact Newton-type methods

We establish a new semilocal convergence results for Inexact Newton-type methods for approximating a locally unique solution of a nonlinear equation in a Banach spaces setting. We show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and [31]. Special cases and numerical examples are also provided in this study.     

理想的Riesz扩张

本在有单位元e的交换Banach代数B中定义了其闭理想A的Riesz扩张R，并证明了R＝A＋Q的充分条件为A={a,a∈A}在局部凸拓扑{|ρ（x)|:ρ∈Ω}下闭，其中Q为B的根基，x为x（∈B）在商映射θ：B→B/Q下的像，Ω为B的谱空间，ρ(x)=ρ(x),↓Ax∈B,ρ∈Ω。     

On the ideal centre of the space of vector valued integrable functions

Let E be a Banach lattice and L¹(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L¹(μ, E) are given. It is shown that L_s^∞(μ, Z(E)) is the ideal centre of L¹(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.      

On almost sure convergence of amarts and martingales without the Radon-Nikodym property

It is shown here that for any Banach spaceE-valued amart (X _n) of classB, almost sure convergence off(X_n) tof(X) for eachf in a total subset ofE ^* implies scalar convergence toX.     

统计测度与N上纯有限可加测度空间的表示定理

证明了N上纯有限可加测度空间的单位球正是统计测度的绝对凸包。     

A new semi-local convergence theorem for the inexact Newton methods

We present a new semi-local convergence theorem for the inexact Newton methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. As special cases of our main result we re-obtain some well-known convergence theorems for Newton methods.     

Weaker conditions for the convergence of Newton’s method

Newton’s method is often used for solving nonlinear equations. In this paper, we show that Newton’s method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Pták (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

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).     

Local convergence of Newton’s method under majorant condition

A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications.