On power series statistical convergence and new uniform integrability of double sequences

In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.

ERRATA TO: “λ-Statistical Convergence of Order α” Acta Mathematica Scientia 2011, 31B(3): 953-959

The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = （λn）. Because this sequences are just the sequences x = （xk） = （k） and x = （xk） = （1） respectively and any term of these sequences can not be 0. In this short not we give Example 3＊ and Example 4＊ to show that the inclusions given in Theorem 2.4 and Theorem 2.9 are strict for some λ = （λn） , α and β such that 0 α β ≤ 1.

SOME NEW TYPE OF DIFFERENCE SEQUENCE SPACES DEFINED BY MODULUS FUNCTION AND STATISTICAL CONVERGENCE

In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.

ON LACUNARY STATISTICAL CONVERGENCE OF ORDER α

In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.

λ-STATISTICAL CONVERGENCE OF ORDER α

In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong（V,λ）-summability of order α are given.

Weighted (<i>λ</i>, <i>μ</i>)-Ideal Statistical Convergence and Strongly Weighted (<i>λ</i>, <i>μ</i>)-Ideal Convergence of Double Sequences of Fuzzy Numbers

The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted (λ, μ)-ideal statistical convergence and strongly weighted (λ, μ)-ideal convergence of double sequences of fuzzy numbers. Also, some properties of these new sequence spaces are investigated.

Lacunary Invariant Statistical Convergence Of Sequences Of Sets With Respect To A Modulus Function

In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.

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.

Statistical Convergence and Measure Convergence Generated by a Single Statistical Measure

The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l∞/l∞ （Iμ）. We also show that the usual sequential convergence is not equivalent to a single measure convergence.

Generalized Statistical Convergence and Statistical Core of Double Sequences

In this paper we extend the notion of A-statistical convergence to the （λ,μ）statistical convergence for double sequences x =（xjk）. We also determine some matrix transformations and establish some core theorems related to our new space of double sequences Sλ,μ.

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.

About Upper and Lower Strong Fractional Weighted Mean Convergence by Moduli for Triple Sequences

In article, I present a study on upper and lower statistical convergence, and upper and lower strong fractional weighted mean convergence by moduli for triple sequences. One of the generalizations of the discrete operator Cesàro, was weighted mean operators, which are linear operators, too. Given a modulus function f, I established that a triple sequence that is f-upper or lower strong fractional weighted mean convergent, in some supplementary conditions, is also f-lower or upper statistically convergent. The results of this paper adapt the results obtained in [1] and [2] to upper and lower strong fractional weighted mean convergence and to triple sequence concept. Furthermore, new concepts can be applied to the approximation theory, topology, Fourier analysis, analysis interdisciplinary with applications electrical engineering, robotics and other domains.

ON WEAK FILTER CONVERGENCE AND THE RADON-RIESZ TYPE THEOREM

The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.

Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions

In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that x = {xijk} is ideal statistically pre-Cauchy if and only if D (xijk, xpqr) ≥ ε, i ≤ m, j ≤ n, t ≤ k}| ≥ δ} ∈I. At the same time, we have proved x = {xijk} is ideal statistically convergent to x0 if and only if . Also, some properties of these new sequence spaces are investigated.

Statistically Convergent Difference Double Sequence Spaces

In this article we define the notion of statistically convergent difference double sequence spaces. We examine the spaces 2l∞（△, q）, 2c（△, q）, 2cB（△, q）, 2cR（△, q）, 2cBR（△,q） etc. being symmetric, solid, monotone, etc. We prove some inclusion results too.

More about the Kernel Convergence and the Ideal Convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.

On Ideal Convergence of Double Sequences in Probabilistic Normed Spaces

The notion of ideal convergence is a generalization of statistical convecgence which has been intensively investigated in last few years. For an admissible ideal ∮ C 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.