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Riesz Triple Almost Lacunary χ3 Sequence Spaces Defined by a Orlicz Function-II
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Riesz Triple Almost Lacunary χ3 Sequence Spaces Defined by a Orlicz Function-II

Vandana1, Deepmala2, Subramanian N3 and Lakshmi Narayan Mishra4,5*
1Department of Management Studies, Indian Institute of Technology, Madras, Chennai 600 036, India
2Mathematics Discipline, PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur 482 005, Madhya Pradesh, India
3Department of Mathematics, SASTRA University, Thanjavur-613 401, India
4Department of Mathematics, Lovely Professional University, Jalandhar Delhi G.T. Road, Phagwara, Punjab 144 411, India
5L. 1627 Awadh Puri Colony Beniganj, Phase III, Opposite Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad-224 001, Uttar Pradesh, India
*Corresponding Author: Lakshmi Narayan Mishra, Department of Mathematics, Lovely Professional University, Jalandhar Delhi G.T. Road, Phagwara, Punjab 144 411, India, Tel: +91 9838375431, Email: [email protected]

Received Date: Oct 05, 2017 / Accepted Date: Dec 12, 2017 / Published Date: Dec 22, 2017

Abstract

The aim of this paper is to introduce a new concept for strong almost Pringsheim convergence with respect to an Orlicz function, combining with Riesz mean for triple sequences and a triple lacunary sequence. We also introduce and study statistics convergence of Riesz almost lacunary χ3 sequence spaces and also some inclusion theorems are discussed.

Keywords: Analytic sequence; Modulus function; Double sequences; Chi sequence; Riesz space; Riesz convergence; Pringsheim convergence

2010 Mathematics subject classification

40A05,40C05,40D05

Introduction

Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w3 for the set of all complex triple sequences (xmnk), where Equation, the set of positive integers. Then, w3 is a linear space under the coordinate wise addition and scalar multiplication.

We can represent triple sequences by matrix. In case of double sequences we write in the form of a square. In the case of a triple sequence it will be in the form of a box in three dimensional case.

Some initial work on double series is found in Apostol [1] and double sequence spaces is found in Hardy, Subramanian et al. [2-9], and many others. Later on investigated by some initial work on triple sequence spaces is found in Sahiner et al. [10], Esi et al. [11-15], Subramanian et al. [16-25] and many others. Some interesting results in this direction can be seen [26-29].

Let (xmnk) be a triple sequence of real or complex numbers. Then the series Equation is called a triple series. The triple series Equation give one space is said to be convergent if and only if the triple sequence (Smnk)is convergent, where,

Equation

A sequence x=(xmnk) is said to be triple analytic if,

Equation

The vector space of all triple analytic sequences are usually denoted by 3. A sequence x=(xmnk) is called triple entire sequence if,

Equation

The vector space of all triple entire sequences are usually denoted by Γ3. Let the set of sequences with this property be denoted by Λ3 and Γ3 is a metric space with the metric,

Equation (1)

forall x=(xmnk) and y=(ymnk) in Γ3. Let φ={finite sequences}.

Consider a triple sequence x=(xmnk). The (m,n,k)th section x[m,n,k] of the sequence is defined by Equation,

Equation

with 1 in the (m,n,k)th position and zero otherwise.

A sequence x=(xmnk) is called triple gai sequence if Equation as m,n,k∞. The triple gai sequences will be denoted by χ3.

Definitions and Preliminaries

A triple sequence x=(xmnk) has limit 0 (denoted by Plimx=0)

(i.e) Equation as m,n,k→∞. We shall write more briefly as P−convergent to 0.

Definition

A modulus function was introduced by Nakano [30]. We recall that a modulus f is a function from [0,∞)[0,∞), such that,

(1) f(x)=0 if and only if x=0

(2) f(x+y)≤ f(x)+f(y), for all x≥0, y≥0,

(3) f is increasing,

(4) f is continuous from the right at 0. Since |f(x)−f(y)≤ f(|xy|), it follows from here that f is continuous on [0,∞).

Definition

Let Equation be sequences of positive numbers and

Equation

Equation

Equation

Then the transformation is given by:

Equation is called the Riesz mean of triple sequence x=(xmnk). If Equation, then the sequence x=(xmnk) is said to be Riesz convergent to 0. If x=(xmnk) is Riesz convergent to 0, then we write PRlimx=0.

Definition

The triple sequence θi,l,j={(mi,nl,kj)} is called triple lacunary if there exist three increasing sequences of integers such that

Equation

Let Equation, and θi,l,j is determine by

Equation

Using the notations of lacunary sequence and Riesz mean for triple sequences.

θi,l,j={(mi,nl,kj)} be a triple lacunary sequence and Equation be sequences of positive real numbers such that

Equation

If the Riesz transformation of triple sequences is RH-regular, and Equation as Equation asEquation as j∞, then Equation is a triple lacunary sequence. If the assumptions Qr∞. as r∞, Equation as s∞ and Equation as t→∞. may be not enough to obtain the conditions Hi→∞. as i→∞, Equation as l∞ and Equation as j→∞ respectively. For any lacunary sequences (mi),(nl) and (kj) are integers.

Throughout the paper, we assume that

Equation

such that Equation asEquation as l∞ and Equation as j∞.

Let

Equation

and Equation and Equation.

If we take Equation for all m,n and k then Equation reduce to Equation.

Let f be an Orlicz function and s=(pmnk) be any factorable triple sequence of strictly positive real numbers, we define the following sequence spaces:

Equation

uniformly in i,l and j.

Equation

uniformly in i,l and j.

Let f be an Orlicz function, p=pmnk be any factorable double sequence of strictly positive real numbers and and Equation be sequences of positive numbers and Equation and Equation,

If we choose Equation for all m,n and k, then we obtain the following sequence spaces.

Equation

uniformly in i,l and j.

Equation

uniformly in i,l and j.

Main Results

Theorem

If f be any Orlicz function and a bounded factorable positive triple number sequence pmnk then Equation is linear space.

Proof: The proof is easy. Therefore omit the proof.

Theorem

For any Orlicz function f, we have Equation

Proof: Let Equation so that for each i,l and j.

Equation

uniformly in i,l and j.

Since f is continuous at zero, for ε>0 and choose δ with 0<δ<1 such that f(t)<ε for every t with 0≤. We obtain the following,

Equation

Hence i,l and j goes to infinity, we are granted Equation.

Theorem

Let θi,l,j={mi,nl,kj} be a triple lacunary sequence and Equation withEquation then for any Orlicz function Equation

Proof: Suppose Equation then there exists δ>0, such that Equation. This impliesEquation Then forEquation we can write for each i,l and j.

Equation

Since Equation, the last three terms tend to zero uniformly in m,n,k in the sense, thus, for each i,l and j:

Equation

Since Equation we are granted for each i,l and j the following:

Equation

The terms

Equation

and

Equation

are both gai sequences for all r,s and u. Thus Ailj is a gai sequence for each i,l and j. Hence Equation.

Theorem

Let θi,l,j={mi,nl,kj} be a triple lacunary sequence and Equation withEquation and Equation then for any Orlicz function Equation.

Proof: Since Equation there exists H>0 such that Equation H for all i,l and j. Let Equation and ε>0. Then there exist i0>0, l0>0 and j0>0 such that for every ai0, bl0 and cj0 and for all i,l and j.

Equation

Equation

Let Equation and p,r and t be such that Equation. Thus we obtain the following:

Equation

Since Equation as i,l,j→∞ approaches infinity, it follows that

Equation

Corollary

Let θi,l,j={mi,nl,kj} be a triple lacunary sequence and Equation be sequences of positive numbers. If Equation, then for any Orlicz function f , Equation.

Definition

Let θi,l,j={mi,nl,kj} be a triple lacunary sequence. The triple number sequence x is said to beEquation convergent to 0 provided that for every ε >0,

Equation

In this case we write Equation

Theorem

Let θi,l,j={mi,nl,kj} be a triple lacunary sequence. If Equation, then the inclusion Equation is strict andEquation

Proof: Let

Equation (2)

Suppose that Equation. Then for each i,l and j

Equation

Since

Equation

for all i,l and j, we get

Equation for each i,l and j. This implies that Equation.

To show that this inclusion is strict, let x=(xmnk) be defined as

Equation

and Equation for all m,n and k. Clearly, x is unbounded sequence. For ε>0 and for all i,l and j we have

Equation

Therefore Equation with the Plim=0. Also note that

Equation

Hence Equation.

Theorem

Let Equation. If the following conditions hold, thenEquation and Equation.

Equation

Proof: Let x=(xmnk) be strongly Equation - almost P convergent to the limit 0. Since

Equation

for (1) and (2), for all i,l and j, we have

Equation

where Equation is as mentioned above. Taking limit i,l,j→∞ in both sides of the above inequality, we conclude that Equation.

Definition

A triple sequence x=(xmnk) is said to be Riesz lacunary of χ almost P− convergent 0 if Equation, uniformly in i,l and j, where Equation.

Definition

A triple sequence (xmnk) is said to be Riesz lacunary χ almost statistically summable to 0 if for every ε>0 the set

Equation has triple natural density zero, (i.e) δ3(Ks)=0. In this we write Equation. That is, for every ε>0,

Equation, uniformly in i,l and j.

Theorem

Let Equation. andEquation for allEquation and for each i,l and j. Let Equation.

Let Equation.

Then,

Equation

for each i,l and j, which implies that Equation uniformly i,l and j. Hence, Equation uniformly in i,l, j. Hence Equation

Conclusion

To see that the converse is not true, consider the triple lacunary sequence Equation for all m,n and k, and the triple sequence x=(xmnk) defined by Equation for all m,n and k.

References

Citation: Vandana, Deepmala, Subramanian N, Mishra LN (2017) Riesz Triple Almost Lacunary χ3 Sequence Spaces Defined by a Orlicz Function-II. J Generalized Lie Theory Appl 11: 285. DOI: 10.4172/1736-4337.1000285

Copyright: © 2017 Vandana, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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