40A05,40C05,40D05

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 (x_mnk), where , 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 (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_mnk)is convergent, where,

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 Γ³. Let the set of sequences with this property be denoted by Λ³ and Γ³ is a metric space with the metric,

(1)

forall x=(x_mnk) and y=(y_mnk) in Γ³. 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 χ³.

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_mnk) is said to be Riesz convergent to 0. If x=(x_mnk) is Riesz convergent to 0, then we write P_R−limx=0.

Definition

The triple sequence θ_i,l,j={(m_i,n_l,k_j)} is called triple lacunary if there exist three increasing sequences of integers such that

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_i,n_l,k_j)} be a triple lacunary sequence and be sequences of positive real numbers such that

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_i→∞. as i→∞, as l∞ and as j→∞ respectively. For any lacunary sequences (m_i),(n_l) and (_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.

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_i,n_l,k_j} be a triple lacunary sequence and with then for any Orlicz function

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 ai₀, b≥l₀ and c≥j₀ 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_i,n_l,k_j} be a triple lacunary sequence and be sequences of positive numbers. If , then for any Orlicz function f , .

Definition

Let θ_i,l,j={m_i,n_l,k_j} be a triple lacunary sequence. The triple number sequence 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_i,n_l,k_j} be a triple lacunary sequence. If , then the inclusion is strict and

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

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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