# Riesz Triple Almost Lacunary χ^{3} Sequence Spaces Defined by a Orlicz Function-II

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**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]

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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 χ^{3} 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 *w*^{3} for the set of all complex triple sequences (*x _{mnk}*), where , the set of positive integers. Then,

*w*

^{3}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 (*x _{mnk}*) be a triple sequence of real or complex numbers. Then the series is called a triple series. The triple series give one space is said to be convergent if and only if the triple sequence (

*S*)is convergent, where,

_{mnk}A sequence *x*=(*x _{mnk}*) is said to be triple analytic if,

The vector space of all triple analytic sequences are usually denoted by ^{3}. A sequence *x*=(*x _{mnk}*) is called triple entire sequence if,

The vector space of all triple entire sequences are usually denoted by Γ^{3}. Let the set of sequences with this property be denoted by Λ^{3} and Γ^{3} is a metric space with the metric,

(1)

forall *x*=(*x _{mnk}*) and

*y*=(

*y*) in Γ

_{mnk}^{3}. Let

*φ*={

*finite sequences*}.

Consider a triple sequence *x*=(*x _{mnk}*). The (

*m*,

*n*,

*k*)

^{th}section

*x*

^{[m,n,k]}of the sequence is defined by ,

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

A sequence *x*=(*x _{mnk}*) is called triple gai sequence if as

*m*,

*n*,

*k*∞. The triple gai sequences will be denoted by χ

^{3}.

#### Definitions and Preliminaries

A triple sequence *x*=(*x _{mnk}*) has limit 0 (denoted by

*Plimx*=0)

(i.e) 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 be sequences of positive numbers and

Then the transformation is given by:

is called the Riesz mean of triple sequence *x*=(*x _{mnk}*). If , then the sequence

*x*=(

*x*) is said to be Riesz convergent to 0. If

_{mnk}*x*=(

*x*) is Riesz convergent to 0, then we write

_{mnk}*P*−

_{R}*limx*=0.

**Definition**

The triple sequence *θ _{i,l,j}*={(

*m*)} is called triple lacunary if there exist three increasing sequences of integers such that

_{i},n_{l},k_{j}Let , and *θ _{i,l,j}* is determine by

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

*θ _{i,l,j}*={(

*m*)} be a triple lacunary sequence and be sequences of positive real numbers such that

_{i},n_{l},k_{j}If the Riesz transformation of triple sequences is RH-regular, and as as as *j*∞, then is a triple lacunary sequence. If the assumptions *Q _{r}*∞. as

*r*∞, as

*s*∞ and as

*t*→∞. may be not enough to obtain the conditions

*H*→∞. as

_{i}*i*→∞, as

*l*∞ and as

*j*→∞ respectively. For any lacunary sequences (

*m*),(

_{i}*n*) and (

_{l}

_{kj}_{)}are integers.

Throughout the paper, we assume that

such that as as *l*∞ and as *j*∞.

Let

and and .

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

Let *f* be an Orlicz function and *s*=(*p _{mnk}*) be any factorable triple sequence of strictly positive real numbers, we define the following sequence spaces:

uniformly in *i*,*l* and *j*.

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 be sequences of positive numbers and and ,

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

uniformly in *i*,*l* and *j*.

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 is linear space.

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

**Theorem**

For any Orlicz function *f*, we have

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

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≤*tδ*. We obtain the following,

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

**Theorem**

Let *θ _{i,l,j}*={

*m*} be a triple lacunary sequence and with then for any Orlicz function

_{i},n_{l},k_{j}**Proof:** Suppose then there exists δ>0, such that . This implies Then for we can write for each *i*,*l* and *j*.

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

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

The terms

and

are both gai sequences for all *r*,*s* and *u*. Thus *A _{ilj}* is a gai sequence for each

*i*,

*l*and

*j*. Hence .

**Theorem**

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

**Proof:** Since there exists *H*>0 such that *H* for all *i*,*l* and *j*. Let and ε>0. Then there exist i0>0, l0>0 and j0>0 such that for every *ai*_{0}, *b*≥*l*_{0} and *c*≥*j*_{0} and for all *i*,*l* and *j*.

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

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

**Corollary**

Let *θ _{i,l,j}*={

*m*} be a triple lacunary sequence and be sequences of positive numbers. If , then for any Orlicz function

_{i},n_{l},k_{j}*f*, .

**Definition**

Let *θ _{i,l,j}*={

*m*} be a triple lacunary sequence. The triple number sequence

_{i},n_{l},k_{j}*x*is said to be convergent to 0 provided that for every

*ε*>0,

In this case we write

**Theorem**

Let *θ _{i,l,j}*={

*m*} be a triple lacunary sequence. If , then the inclusion is strict and

_{i},n_{l},k_{j}**Proof:** Let

(2)

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

Since

for all *i*,*l* and *j*, we get

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

To show that this inclusion is strict, let *x*=(*x _{mnk}*) be defined as

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

Therefore with the *P*−*lim*=0. Also note that

Hence .

**Theorem**

Let . If the following conditions hold, then and .

Proof: Let *x*=(*x _{mnk}*) be strongly - almost

*P*convergent to the limit 0. Since

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

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

**Definition**

A triple sequence *x*=(*x _{mnk}*) is said to be Riesz lacunary of

*χ*almost

*P*− convergent 0 if , uniformly in

*i*,

*l*and

*j*, where .

**Definition**

A triple sequence (*x _{mnk}*) is said to be Riesz lacunary

*χ*almost statistically summable to 0 if for every

*ε*>0 the set

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

, uniformly in *i*,*l* and *j*.

**Theorem**

Let . and for all and for each *i*,*l* and *j*. Let .

Let .

Then,

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

#### Conclusion

To see that the converse is not true, consider the triple lacunary sequence for all *m*,*n* and *k*, and the triple sequence *x*=(*x _{mnk}*) defined by for all

*m*,

*n*and

*k*.

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Citation:
Vandana, Deepmala, Subramanian N, Mishra LN (2017) Riesz Triple Almost Lacunary χ^{3} 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.