Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Weak Orlicz space and its convergence theorems

In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.     

WEAK ORLICZ SPACE AND ITS CONVERGENCE THEOREMS

In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.     

CONVERGENCE BALL OF ITERATIONS WITH ONE PARAMETER

§1Introduction OfthevariousiterationmethodsusedtosolvetheequationF(x)=0,thefamilyof iterativemethodswithoneparameter xn+1=xn-I+12PF(I-αPF)-1F′(xn)-1F(xn),PF(x)=F′(x)-1F″(x)F′(x)-1F(x),α∈[0,1],(1)isremarkablebecauseitincludesthesuper-Halleymethod,theChebyshevmethodand Halley'smethodasitsspecialcaseswhenαisequalto1,0and1/2,respectively.Thereareanumberofpapersconcerningtheabovethreeiterations.In[1-7],Halley's methodisstudied.Amongthesereferences,[1]givestheconvergenceofHalley'sme…     

一类非齐次树上二重马氏链的若干强偏差定理

强偏差定理一直是概率论研究的中心问题之一.本文通过构造一非负鞅,利用Doob鞅收敛定理得到了一类特殊非齐次树上二重马尔可夫链关于三元状态序组出现频率的若干强偏差定理.     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.     

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.     

Some Convergence Properties for ψ-Mixing Sequences

In this paper,we extend the Kolmogorov-type inequality to the case of ψ-mixing sequences.Moreover,we study the strong limit theorems for partial sums of ψ-mixing random variables.As a result,we extend the Khintchine-Kolmogorov-type convergence theorem,the three series theorem,Marcinkiewicz strong law of large number to the case of ψ-mixing sequences.     

$\psi$混合序列的一些收敛性质

In this paper,we extend the Kolmogorov-type inequality to the case of ψ-mixing sequences.Moreover,we study the strong limit theorems for partial sums of ψ-mixing random variables.As a result,we extend the Khintchine-Kolmogorov-type convergence theorem,the three series theorem,Marcinkiewicz strong law of large number to the case of ψ-mixing sequences.     

可测函数序列关于弱收敛概率测度序列积分的极限定理

研究了可测函数序列关于弱收敛概率测度序列积分的极限定理及其控制收敛定理,并给出了概率测度弱收敛的若干新的等价条件.     

Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub’s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.     

ON THE CONVERGENCE OF PADE-TYPE APPROXIMANTS IN TWO VARIABLES

In [1], C. Brezinski raised three unsolved questions in studing multivariable Pade-iype approximation, and the convergence of multivariable Pade-iype approximants under the general conditions is one of them. We have only seen in [2] that the convergence of the mullivariable Fade-type approximants is discussed for a kind of functions defined by Stieltjes integrations under some special conditions. However, the general convergence theorem has not been seen so far. In this paper, some error formulae of bivariate Fade-type approximants in in-legral form are given. By virtue of them, a number of convergence theorems of bivariate Pade-iype approximants are proved under general conditions.     

一类并行数值Schwarz格式的收敛性及比较定理

本文对［１］给出的一类求解椭圆型偏微分方程的并行数值Ｓｃｈｗａｒｚ格式作了进一步的分析，给出了一个新的收敛定理，根据该收敛定理，可以把文［１］中的主要收敛快慢进行了比较，得出了一些比较结果。     

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.     

Gauss-Newton法的收敛球与解的唯一性球

0　引　言本文研究非线性最小二乘问题min F( x)∶ =12 f( x) Tf ( x) ( EP)的 Gauss-Newton法的局部收敛性 ,其中 f:Rn→ Rm是 Frechet可微的 ,m≥ n.非线性最小二乘问题在数据拟合 ,参数估计和函数逼近等方面有广泛的应用 .在工程应用中也起到很大作用 ,例如在神经网络中 ,对小波问题 ,FP网络等方面的数据 (图形 )传输 ,数据 (图形 )压缩等方面有极其重要的理论和实际意义 .目前 ,求解最小二乘问题的最基本的方法之一是 Gauss-Newton法 [1 ]xn+1 =xn -[f′( xn) Tf′( x) ] - 1 f′( xn) Tf( xn) . ( GN)就我们所知 ,目前关于 Gau…     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

任意随机序列的变换及其a.s.收敛性

本文利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，研究随机变量序列变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。     

B值随机变量序列的局部收敛定理

本文的目的是研究B值随机变量序列的局部收敛定理。作为推论，得到了一类B值鞅差序列的极限定理和若干经典的独立随机变量序列的极限定理。     

关于树上连续状态马氏泛函的一类强偏差定理

强偏差定理一直是国际概率论界研究的中心课题之一.通过构造适当的非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,给出了一类特殊非齐次树上连续状态马氏泛函的若干强偏差定理.     

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.