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The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.

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The idea of [λ, μ] -almost convergence (briefly, F[λ, μ] -convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on F[λ, μ] such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform F[λ, μ]-convergence of double sequences x = (xjk) into F[λ, μ]-convergence. We also define a F[λ, μ]-core of x = (xjk) and determine a Tauberian condition for core inclusions and core equivalence.

The concept of statistical convergence of a sequence was first intro-duced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix A in place of C\ . The main result in this paper is a theorem that gives meaning to the state-ment: S= {sn} converges to L statistically (T) if and only if "most" of the subsequences of 5 converge, in the ordinary sense, to L . Here T is a regular, nonnegative and triangular matrix. Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented.

The idea of statistical convergence was first introduced by H. Fast [Colloq. Math. 2, 241-244 (1951; Zbl 0044.33605)] but the rapid developments were started after the papers of J. A. Šalát [Math. Slovaca 30, 139-150 (1980; Zbl 0437.40003)] and J. A. Fridy [Analysis 5, 301-313 (1985; Zbl 0588.40001)]. Nowadays it has become one of the most active areas of research in the field of summability. In this paper we define and study statistical analogue of convergence and Cauchy for double sequences. We also establish the relation between statistical convergence and strongly Cesàro summable double sequences.

In this paper, we discuss almost convergent sequences. The concept of almost convergence was first introduced by G. G. Lorentz in 1948. Since then, various types of generalizations for almost convergence have been constructed and studied. Our concern is also to generalize the almost convergence. We construct a generalized almost convergence (GAC) in a natural way through the use of Lorentz-type definition, and give an analogue of Lorentz’s result. It should be noted that the definition of our GAC does not require normed space nor boundedness of sequences, although Lorentz-type definitions usually require it. On the other hand, we also show that our GAC eventually has a certain type of boundedness in spite of its definition.

Consider the Banach space m of real bounded sequences, x, with (Formula presented.). A positive linear functional L on m is called an S-limit if (Formula presented.) for every characteristic sequence (Formula presented.) of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.

The sequence X is statistically convergent to L if for each ∊ > 0, limn n-1 [the number of k ≤ n : | xk - L | ≥ ∊] = 0; x is a statistically Cauchy sequence if for each ∊ > 0 there is a positive integer N = N(e) such that [formula omitted][the number of k ≤ n : | xn - xk | ≥ ∊] = 0. n These concepts are shown to be equivalent. Also, statistical convergence is studied as a regular summability method, and it is shown that it cannot be included by any matrix method. There are two Tauberian theorems proved: one uses the Tauberian condition Δxk = 0(1/k), which is shown to be “best possible,” and the other is concerned with gap sequences.

The notion of the statistical convergence of sequences of real numbers was introduced in papers [1] and [5]. In the present paper we shall show that the set of all bounded statistically convergent sequences of real numbers is a nowhere dense subset of the linear normed space m (with the sup-norm) of all bounded sequences of real numbers and the set of all statistically convergent sequences of real numbers is a dense subset of the first Baire category in the Frechet space s.

The sequence x is said to be Sθ-convergent to L if for each ϵ > 0, limr (1/hr){the number of k ∈ Ir: |xk − L| ≥ ϵ} = 0, where θ = {kr} is an increasing sequnece of integers such that k0 = 0, hr ≔ kr − kr − 1 → ∞ as r → ∞ and Ir ≔ (kr − 1, kr]. In this paper we define the Sθ-analog of the Cauchy criterion for convergence and show that tt is equivalent to Sθ-convergence. Also, Sθ-convergence is compared to other summability methods, and it is shown that the Sθ method can not be included by any matrix method. In addition, a Tauberian theorem for Sθ-convergence is given.

Following the concept of statistical convergence and statistical cluster points of a sequence x, we give a denition of statistical limit supe- rior and inferior which yields natural relationships among these ideas: e.g., x is statistically convergent if and only if st-liminfx = st-limsupx .T he statistical core of x is also introduced, for which an analogue of Knopp's Core Theorem is proved. Also, it is proved that a bounded sequence that is C1-summable to its statistical limit superior is statistically convergent.

In this paper we consider the idea of $I$ - convergence of nets in a topological space and derive several basic properties. This idea extends the concept of $I$ - convergence of sequences considered so far in various spaces.

Following the concept of a statistically convergent sequence x x , we define a statistical limit point of x x as a number λ \lambda that is the limit of a subsequence { x k ( j ) } \{ {x_{k(j)}}\} of x x such that the set { k ( j ) : j ∈ N } \{ k(j):j \in \mathbb {N}\} does not have density zero. Similarly, a statistical cluster point of x x is a number γ \gamma such that for every ε > 0 \varepsilon > 0 the set { k ∈ N : | x k − γ | > ε } \{ k \in \mathbb {N}:|{x_k} - \gamma | > \varepsilon \} does not have density zero. These concepts, which are not equivalent, are compared to the usual concept of limit point of a sequence. Statistical analogues of limit point results are obtained. For example, if x x is a bounded sequence then x x has a statistical cluster point but not necessarily a statistical limit point. Also, if the set M := { k ∈ N : x k > x k + 1 } M: = \{ k \in \mathbb {N}:{x_k} > {x_{k + 1}}\} has density one and x x is bounded on M M , then x x is statistically convergent.

It is shown that if a sequence is strongly p-Cesàro summable or w p convergent for 0<p<∞ then the sequence must be statistically convergent and that a bounded statistically convergent sequence must be w p convergent for any p, 0<p<∞. It is also shown that the statistically convergent sequences do not form a locally convex FK space. A characterization of conservative matrices which map the bounded statistically convergent sequences into convergent sequences is given and applied to Nörlund and Nörlund-type means.

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.(Received July 29 1987)(Revised September 30 1987)

A double sequence x = { x jk : j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if
that is, the average value of { x jk } taken over any rectangle {( j, k ): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n . The notion of almost convergence for single sequences was introduced by Lorentz [ 1 ].

There are two well-known non-matrix summability methods which we will consider, namely “almost convergence” and “statistical
convergence”. The results presented in this paper will be of two types, dealing with Lebesgue measure and Baire category.
Establishing a one-to-one correspondence between the interval (0; 1] and the collection of all subsequences of a given sequence
s = (s
n), we will examine the measure and category of the set of all almost convergent subsequences of (s
n). Similar questions for statistical and lacunary statistical convergence are considered. Results on rearrangements of sequences
are also presented.

Thorie des operations liniaries

- S Banach

Banach, S., Thorie des operations liniaries, Warszawa, 1932.

- P Kostyrko
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- W Wilczynski

Kostyrko, P., alat, T. and Wilczynski, W., I-convergence. Real Anal. Exch. 26(2) (2001) 669-686.