Lacunary Convergence Research Articles

STRONGLY SUMMABLE AND STATISTICALLY CONVERGENT FUNCTIONS

Strongly summable sequences and lacunary strongly summable sequences were studied by several authors including [5]. Also statistically convergent sequences and lacunary statistical convergent sequences were studied several authors, including [2],[3],[6],[7].In this paper instead of sequences, by taking real valued functions x,measurable (in the Lebesque sense) in the interval (1;1), we have given definitions of summability, strong summability, lacunary convergence, lacunary strong convergence, strongly almost convergence, statistical convergence and lacunary statistical convergence of these functions. Also we have given some inclusion relations.

Results on lacunary difference ideal convergence in intuitionistic fuzzy normed linear spaces

The concept of lacunary ideal convergence in intuitionistic fuzzy normed linear space IFNLS was introduced by the present author [P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 2012, 708-715]. In the current paper, the notion of lacunary difference ideal convergence LDIC has been introduced. Some previous results have been improved and extended in this new setting and some new results are incorporated as well. An open problem, discussed in the above mentioned paper, has also been solved here.

Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space

The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.

Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong (A)-convergence, whereA=(aik)is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

STATISTICAL AND LACUNARY CONVERGENCE-ITS LATEST DEVELOPMENT FROM THE POINT OF VIEW OF SEQUENCE SPACES

The development of the theory of sequence spaces is nowadays effected by the introduction of new convergence methods and theories in the process. Some of them are statistical convergence, lacunary convergence, lacunary statistical convergence etc (see [1], [2], [3], [8], [11]). Here, I have basically concentrated on these concepts. Statistical convergence while introduced over nearly fifty years ago has only recently become an area of active research in sequence spaces. Fast [5] extended the concept of sequential limit which he called statistical convergence. Schoenberg (1959) gave some basic properties of statistical convergence and studied the concept as summability method. The basic concept of statistical convergence is based on the notion of natural density of sets A ⊆ N = {1, 2, . . . , n, . . .}, refer to [5], [6]. If A ⊆ N and A(n)=|A ∩ {1, 2, . . . , n}|, where the vertical bar denotes the cardinality of the

The sequence space $E_{n}^{q}\left( M,p,s\right) $ and $ N_{k}-$ lacunary statistical convergence

In this paper we define the sequence space $E_{n}^{q}(M,p,s)$ by using an Orlicz function and we study various properties and obtain some inclusion relations involving this space. We give some relations between $N_{k}-$lacunary statistical convergence and strongly $N_{k}-$lacunary convergence.

A Generalization of Lacunary Equistatistical Convergence of Positive Linear Operators

In this paper we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. In particular we study lacunary equi-statistical convergence of approximating operators on spaces, the spaces of all real valued continuous functions de ned on and satisfying some special conditions.

On Sequence Spaces of Invariant Means and Lacunary Defined by a Sequence of Orlicz Functions

The purpose of this paper is to introduce and study some sequence spaces which are dened by combining the concepts of a sequence of Orlicz functions, invariant mean and lacunary convergence. We establish some inclusion relations between these spaces under some conditions.

On Some Lacunary Generalized Difference Sequence Spaces of Invariant Means De ned by a Sequence of Modulus Function

The aim of this paper is to introduce and study the sequence spaces ( w; �; F; p; q ) 1(∆ m ), ( w; �; F; p; q ) 1(∆ m ) and ( w; �; F; p; q ) 0(∆ m ), which arise from the no- tions of generalized difference sequence space, lacunary convergence, invariant mean and a sequence of Moduli F = ( f k). We establish some inclusion relations between these spaces under some conditions.

Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces

The aim of this article is to introduce and study the notion of lacunary ideal convergence in intuitionistic fuzzy normed linear spaces as a variant of the notion of ideal convergence. Also, a new concept, called lacunary convergence, is introduced. Using these two notions, lacunary I -limit points and lacunary I -cluster points have been defined and the relation between them has been established. Characterization for lacunary ideal convergence preserving linear operators has been given. Furthermore, the notions of lacunary Cauchy and lacunary I -Cauchy sequences are introduced and studied.

Lacunary equi-statistical convergence of positive linear operators

Abstract 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.

Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers

The purpose of this paper is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of sequences of fuzzy numbers. We give some relations related to these concepts. We establish some connections between strongly almost lacunary convergence and almost lacunary statistical convergence of sequences of fuzzy numbers. It is also shown that if a sequence of fuzzy numbers is strongly almost lacunary convergent with respect to an Orlicz function then it is almost lacunary statistical convergent.

On I-lacunary strong convergence in 2-normed spaces

In this article we introduce I- lacunary convergence of sequences in 2-normed spaces. Mathematics Subject Classification: 40A05, 46A45; 46B70

Some generalized difference sequence spaces of invariant meansdefined by Orlicz functions

We define the sequence spaces , , and which are defined by combining the concepts of Orlicz functions, invariant means, and lacunary convergence. We also study some inclusion relations and linearity properties of the above‐mentioned spaces. These are generalizations of those defined and studied by Savaş and Rhoades in 2002 and some others before.

On lacunary invariant sequence spaces defined by a sequence of modulus functions

The purpose of the paper is to introduce and study some sequence spaces which are defined by combining the concepts of lacunary convergence, invariant mean and the sequence of modulus functions. We also examine some topological properties of these spaces.

On some new sequence spaces of invariant means defined by Orlicz functions

The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of a Orlicz function, invariant mean and lacunary convergence. We also examine some topological properties of these spaces and establish some elementary connections between lacunary [w]σ -convergence and lacunary [w]σ -convergence with respect to an Orlicz functions which satisfy a Δ2 -condition. Mathematics subject classification (2000): Primary 46A45, 40H05; Secondary 40A05, 40D25.

Lacunary convergence of series

A simpler proof is given for the recent result of I. Labuda and the author that a series in the space Lo (lambda) is subseries convergent if each of its lacunary subseries converges.

Lacunary convergence of series in $L_0$

Lacunary convergence of series in $L_0$

Decomposition theorem and lacunary convergence of Riesz-Bochner means of Fourier transforms of two variables

Decomposition theorem and lacunary convergence of Riesz-Bochner means of Fourier transforms of two variables