Introduction

The idea of statistical convergence was introduced by Steinhaus [17] and also independently by Fast [2] for real or complex sequences. Statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence.

Let K be a subset of the set of positive integers N x N x N and let us denote the set {(m,n,k) £ K:m < u,n <  v,k < w by K_uvw Then the natural density of K is given by d(K)= lim_uvw ®¥ K|_uvw|/_uvw where |K_uvw| denotes the number of elements in K_uvw. Clearly, a finite subset has natural density zero, and we have d(K^c) = 1-d(K) where K^c = N/K is the complement of K. If K1 Í K2, then d(K₁) < d(K₂).

Throughout the paper, R denotes the real of three dimensional space with metric (X, d). Consider a triple sequence x = (X_mnk) such that (X_mnk) € R, m.n.k € N.

A triple sequence x = (X_mnk) is said to be statistically convergent to 0 € R, written as st - lim x = 0, provided that the set

has natural density zero for any e >0.   In this case,  is called the statistical limit of the triple sequence x.

If a triple sequence is statistically convergent, then for every e >0 infinitely many terms of the sequence may remain outside the e- neighbourhood of the statistical limit, provided that the natural density of the set consisting of the indices of these terms is zero. This is an important property that distinguishes statistical convergence from ordinary convergence. Because the natural density of a finite set is zero, we can say that every ordinary convergent sequence is statistically convergent.

If a triple sequence x = (Xmnk) satisfies some property P for all m.n.k except a set of natural density zero, then we say that the triple sequence x satisfies P for almost all (m,n,k) and we abbreviate this by a.a. (m,n,k).

Let (X_mjnjke) be a sub sequence of x = (Xmnk). If the natural density of the set K = {(mj,nj,ke) €N³: (i,j,J) €N³} is different from zero, then (X_mjnjke) is called a non thin sub sequence of a triple sequence x.

c€R is called a statistical cluster point of a triple sequence x= (X_mnk) provided that the natural density of the set

is different from zero for every e > 0. We denote the set of all statistical cluster points of the sequence x by G x.

A triple sequence x = (x_mnk) is said to be statistically analytic if there exists a positive number M such that

The theory of statistical convergence has been discussed in trigonometric series, summability theory, measure theory, turnpike theory, approximation theory, fuzzy set theory and so on.

The idea of rough convergence was introduced by Phu [8], who also introduced the concepts of rough limit points and roughness degree. The idea of rough convergence occurs very naturally in numerical analysis and has interesting applications. Aytar [1] extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence. Pal et al. [7] extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence.

Let (X,p) be a metric space. For any non empty closed subsets f, f_mnk  Ì X(m,n,k €I_rst) we say that the triple sequence of functions of (fmnk) is wijsman l statistical convergent of order a to f is the triple sequence of functions (d(fmnk,x) is statistically convergent to d(f,x), i.e., for  e>0 and for each fex

In this case, we write St- lim_mnk f_mnk = f of f_mnk →f_(WS). The triple sequence of functions of (f_mnk) is bounded if sup_mnkd(f_mnk,x)<µ for each fÎX.

In this paper, we introduce the notion of Wijsman rough rl^a statistical convergence of order a of triple sequence of functions. Defining the set of Wijsman rough rl^a  statistical convergence of order a limit points of a triple sequence of functions, we obtain to Wijsman rl^a statistical convergence of order a criteria associated with this set. Later, we prove that this set of Wijsman rl^a statistical convergence of order  of cluster points and the set of Wijsman rough rl^astatistical convergence of order  limit points of a triple sequence of functions.

The a- density of a subet E of N. Let a be a real number such that 0 < a < 1. The a- density of a subet E of N is defined by

provided the limit exists, where |{m<r,n<s,k<t:(m,n,k)ÎE}| denotes the number of elements of E not exceeding (rst).

It is clear that any finite subset of N has a zero a density and d_a(E^c)= 1- da(E) does not hold for 0

A triple sequence (real or complex) can be defined as a function x: NXNXN→R(C), where  N,R and C denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by Sahiner et al. [9,10], Esi et al. [2-4], Datta et al. [5],Subramanian et al. [11], Debnath et al. [6], Esi et al. [12], and many others.

  Throughout the paper let r  be a nonnegative real number.

Definitions and Preliminaries

Definition

is called the Wijsman rl^{a -}statistical convergence limit set of the triple sequences of functions.

Definition

A triple sequence of functions (f_mnk) and ^aÎ(o,1]said to be Wijsman rl^a- convergent to the function f denoted by f_mnk  →rl^a f^,  if LIM ^rla  f ¹f. In this case, rl^a is called the Wijsman rl^a convergent to the functions of degree of the triple sequence of functions f = (f_mnk). for r = 0, we get the ordinary convergence.

Definition

holds for every e > 0 and almost all (m,n,k). Here rl^a is called the wijsman rl^a roughness of degree. If we take r= 0, then we obtain the ordinary Wijsman statistical convergence of triple sequence of functions.

In a similar fashion to the idea of classic Wijsman rl^a rough convergence, the idea of Wijsman ​​​​​​rl^a rough statistical convergence to the triple sequence spaces of functions can be interpreted as follows:

Assume that a triple sequence of functions = (f_mnk) is Wijsman rl^a statistically convergent to the functions and cannot be measured or calculated exactly; one has to do with an approximated (or Wijsman rl^astatistically approximated) triple sequence of functions of

Then the triple sequence of functions f_mnk is not rl^a statistically convergent to the functions any more, but as the inclusion

i.e., the triple sequence spaces of functions of f_mnk is Wijsman rl^a statistically convergent to the functions in the sense of definition (2.3)

In general, the Wijsman rough rl^a statistical convergence to the functions of limit of a triple sequence of functions may not unique for the Wijsman roughness degree r > 0. So we have to consider the so called Wijsman rl^a- statistical convergence to the functions of limit set of a triple sequence of functions of (fmnk), which is defined by.

Main Results

Theorem

A triple sequence of functions (f_mnk) and Îa (0,1] be any real number of Wijsman rl^a statistically convergence to the functions, we have (st- LIM^rla​ f_mnk​) <2r. In general dian (st- LIM^rla​ f_mnk​) has an upper bound.

Now let us prove the second part of the theorem. Consider a Wijsman triple sequence of functions of real numbers of (f_mnk) such that st-lim f_{mnk =} f_. Let e > 0. Then we can write

Because the triple sequence of functions of real numbers of f_mnk Because the triple sequence of functions of real numbers of f, we have.

Theorem

is valid.

Theorem

Proposition

If a Wijsman rl^a statistically convergence to the functions of (d(f_{mnk, x})) is analytic, then there exists a non-negative real number r such that  (st- LIM^rla​ f_mnk​ ¹f.

Theorem

A triple sequence of functions (f_mnk) and a Î(o,1)  be any real number of (d(f_mnk,x)) is Wijsman rl^a statistically convergence to the functions of analytic if and only if there exists a non-negative real number r such that st- LIM^rlaf_mnk​ ¹f.

Proof Since the triple sequence of functions of d(f_mnk,x) is Wijsman rl^a statistically convergence to the functions of analytic, there exists a positive real number M such that.

for each e > 0. Then we say that almost all d(f_mnk) are contained in some ball with any radius greater than . So the triple sequence of functions of d (f_mnk,x) is Wijsman rl^a statistically convergence to the functions of analytic.

Remark

If f ' = (fmj,nj,ke) is a sub sequence of functions of (f_mnk), then LIM^rla f_mnk Í LIM^rla f_mnk. But it is not valid for Wijsman ^rla statistical convergence to the functions. For

of real numbers. Then the triple sequence of functions of f' = (1,64.739..) is a sub sequence of functions of f. We have st- LIM^rlaf_mnk​ =f.

Theorem

Let f' = (f_,mj,nj,ke) is a non thin sub sequence of functions of Wijsman ^rla statistically convergence to the functions of f = (f_mnk), then st-LIM^rlaf_mnk Í st -LIM^rlaf_mnk.

Proof: Omitted.

Theorem

Theorem

Theorem

Because (d(f,x))' is a Wijsman ^rla statistical convergence to the functions of cluster point of the triple sequence of functions of d(f_mnk,x) by Theorem (2.4) this inequality implies that d(f,x) Ë st- LIM^{rla (}d(f,x)^). This contradicts the fact |d(f,x)- d(f,x)| = r and st- LIM^{rla (}d(f,x)⁾ = B_rla(d_(f,x)). Therefore, d(f,x). is the unique Wijsman_rla statistical convergence to the functions of cluster point of the triple sequence of functions of d(f_mnk,x). Hence the Wijsman ^rla statistical convergence to the functions of cluster point of a Wijsman ^rla statistically convergence to the functions of analytic is unique, then the triple sequence of functions (df_mnk,x) is wijsman ^rla statistically convergent to the functions of d(f,x).

Theorem

Competing Interests

The authors declare that there is not any conflict of interests regarding the publication of this manuscript.

References

1. S. Aytar 2008. Rough statistical Convergence, Numerical Functional Analysis Optimization, 29(3),  291-303.
2. A. Esi , On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews:Discrete Mathematical Structures, 1(2), (2014),  16-25.
3.  A. Esi and M. Necdet Catalbas,Almost convergence of triple sequences, Global Journal of Mathematical Analysis, 2(1), (2014),  6-10.
4. A. Esi and  E. Savas,  On lacunary statistically convergent triple sequences in probabilistic normed space,Appl.Math.and Inf.Sci., 9 (5) , (2015),  2529-2534.
5. A. J. Dutta, A. Esi and B.C. Tripathy,Statistically convergent triple sequence spaces defined by Orlicz function , Journal of Mathematical Analysis, 4(2), (2013),  16-22.
6. S. Debnath, B. Sarma and B.C. Das ,Some generalized triple sequence spaces of real numbers , Journal of nonlinear analysis and optimization, Vol. 6, No. 1 (2015),  71-79.
7. S.K. Pal, D. Chandra and S. Dutta 2013. Rough ideal Convergence, Hacee. Jounral Mathematics and Statistics, 42(6),  633-640.
8. H.X. Phu 2001. Rough convergence in normed linear spaces, Numerical Functional Analysis Optimization, 22,  201-224.
9. A. Sahiner,  M. Gurdal and F.K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math. , 8 No. (2)(2007),  49-55.
10. A. Sahiner, B.C. Tripathy , Some I  related properties of triple sequences,  Selcuk J. Appl. Math. , 9 No. (2)(2008),  9-18.
11. N. Subramanian and A. Esi, The generalized tripled difference of χ3 sequence spaces, Global Journal of Mathematical Analysis, 3 (2) (2015),  54-60
12. A.Esi, N. Subramanian and A. Esi, Triple rough statistical convergence of sequence of Bernstein operators, Int.j.adv.appl.sci, 4(2)(2017), 28-34.