Wijsman rough λ Statistical convergence of order α of triple Sequence of Functions

In this paper, using the concept of natural density, we introduce the notion of Wijsman rough λ statistical convergence of order α triple sequence of functions. We define the set of Wijsman rough λ statistical convergence of order α of limit points of a triple sequence spaces of functions and obtain Wijsman λ statistical convergence of order α criteria associated with this set. Later, we prove that this set is closed and convex and also examine the relations between the set of Wijsman rough λ statistical convergence of order α of cluster points and the set of Wijsman rough λ statistical convergence of order α limit points of a triple sequences of functions.


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×N×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 where |K uvw | denotes the number of elements in K uvw .Clearly, a finite subset has natural density zero, and we have δ(K c )=1-δ(K) where

Wijsman rough λ Statistical convergence of order α of triple Sequence of Functions
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-limx=0, provided that the set {(m,n,k)∈N 3 :|x_mnk,0|≥δ} has natural density zero for any ε>0.In this case, 0 is called the statistical limit of the triple sequence x.
If a triple sequence is statistically convergent, then for every ε>0, infinitely many terms of the sequence may remain outside the ε-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=(x mnk ) 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 minjkl ) ) be a sub sequence of x=(x mnk ).If the natural density of the set K={(m i ,n j ,k l )∈N 3 :(i,j,l)∈N 3 } is different from zero, then (x minjkl ) 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 {(m,n,k)∈N 3 :|x mnk -c|<ε} is different from zero for every ε>0.We denote the set of all statistical cluster points of the sequence x by Γ 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,ρ) be a metric space.For any non empty closed subsets f,f mnk ⊂X(m,n,k∈I rst ), we say that the triple sequence of functions of (f mnk ) is wijsman λ statistical convergent of order α to f is the triple sequence of functions (d(f mnk ,x) ) is statistically convergent to d(f,x), i.e., for ε>0 and for each f∈X In this case, we write St-lim mnk f mnk =f or f mnk →f(WS).The triple sequence of functions of (f mnk ) is bounded if sup mnk d(f mnk ,x)<∞ for each f∈X.
In this paper, we introduce the notion of Wijsman rough statistical convergence of order α of triple sequence of functions.Defining the set of Wijsman rough rλ α statistical convergence of order α limit points of a triple sequence of functions, we obtain to Wijsman rλ α statistical convergence of order α criteria associated with this set.Later, we prove that this set of Wijsman rλ α statistical convergence of order α of cluster points and the set of Wijsman rough rλ α statistical convergence of order α limit points of a triple sequence of functions .
The α-density of a subet E of N. Let α be a real number such that 0<α≤1.The α-density of a subet E of N is defined by δ α (E)=lim rst 1/(rst) α |{m≤r,n≤s,k≤t:(m,n,k)∈E}| p r o v i d e d t h e l i m i t ex i s t s , w h e r e |{m≤r,n≤s,k≤t:(m,n,k)δE}| denotes the number of elements of E not exceeding (rst).
It is clear that any finite subset of N has a zero α density and δ α (E c )=1-δ α (E) does not hold for 0<α<1 in general, the equality holds only if α=1.Note that the α-density of any set reduces to the natural density of the set in case α=1.
A triple sequence (real or complex) can be defined as a function x:N×N×N→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][3][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
A triple sequence of functions (f mnk ) and α∈(0,1] said to be Wijsman rλ α -convergent to the function f denoted by f mnk → rλα f, provided that ∀ε>0 ∃(mε,n ε ,k ε )∈N 3 The set is called the Wijsman rλ α -statistical convergence limit set of the triple sequences of functions.

definition
A triple sequence of functions (f mnk ) and α∈(0,1] said to be Wijsman rλ α -convergent to the function f denoted by f mnk → rλα f, if LIM rλα f ≠ ϕ.In this case, rλ α is called the Wijsman rλ α convergent to the functions of degree of the triple sequence of functions f=(f mnk ).For r=0, we get the ordinary convergence.

definition
A triple sequence of functions (f mnk ) and α∈(0,1] said to be Wijsman rλ α -convergent to the functions f denoted by f mnk → rλα f, provided that the set has natural density zero for every ε>0, or equivalently, if the condition st-limsup|d(f mnk ,x)-d(f,x) |≤r is satisfied.
In addition, we can write f mnk → rλα st f if and only if the inequality lim rst 1/(λ α rst )|{(m,n,k)∈I rst :|d(f mnk ,x)-d(f,x) |<r+ε} |=0 holds for every ε>0 and almost all (m,n,k).Here rλ α is called the wijsman rλ α 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 rλ α rough convergence, the idea of Wijsman rλ α 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 rλ α statistically convergent to the functions and cannot be measured or calculated exactly; one has to do with an approximated (or Wijsman rλ α statistically approximated) triple sequence of functions of f=(f mnk ) satisfying |d(f mnk ,x-y)-d(f,x-y) |≤r for all m,n,k (or for almost all (m,n,k), i.
i.e., the triple sequence spaces of functions of f mnk is Wijsman rλ α -statistically convergent to the functions in the sense of definition (2. 3)   In general, the Wijsman rough rλ α 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 rλ α -statistical convergence to the functions of limit set of a triple sequence of functions of (f mnk ), which is defined by st-LIM(rλ α ) f mnk ={f∈R:f mnk → rλα st) f}.
The triple sequence of functions of f_mnk is said to be Wijsman rλ α -statistically convergent to the functions provided that st-LIM(rλα ) f mnk ≠ϕ.It is clear that if st-LIM(rλ α ) f mnk ≠ϕ for a triple sequence of functions (f mnk ) of real numbers, then we have We know that LIM rλα =ϕ for an unbounded triple sequence of functions of (f mnk ).But such a triple sequence of functions of might be Wijsman rλ α rough statistically convergent to the functions.For instance, define in R. Because the set {1,64,739, ... } has natural density zero, we have and LIM rλα f mnk =ϕ for all r≥0.

theorem
A triple sequence of functions (f mnk ) and α∈(0,1] be any real number of Wijsman rλ α statistically convergence to the functions, we have diam(st-LIM ρλα f mnk )≤2r.In general diam(st-LIM ρλα f mnk ) has an upper bound.

theorem
A triple sequence of functions (f mnk ) and α∈(0,1] be any real number of (d(f mnk ,x) ) is Wijsman rλ α statistically convergence to the functions of analytic if and only if there exists a non-negative real number r such that .

Proof
Since the triple sequence of functions of d(f mnk ,x) is Wijsman rλ α statistically convergence to the functions of analytic, there exists a positive real number M such that Define where Then the set contains the origin of R.So we have .
If for some r≥0, then there exists d(f,x) such that f_mnk, i.e., for each ε>0.Then we say that almost all d(f mnk ,x) are contained in some ball with any radius greater than r.So the triple sequence of functions of d(f mnk ,x) is Wijsman rλ α statistically convergence to the functions of analytic.

remark
If i s a s u b s e q u e n c e of functions of (f mnk ), then .But it is not valid for Wijsman rλ α statistical convergence to the functions.For Example: Define of real numbers.Then the triple sequence of functions of f'=(1,64,739,...) is a sub sequence of functions of f.We have and .
theorem Let is a non thin sub sequence of functions of Wijsman rλ α statistically convergence to the functions of f=(f mnk ), then Proof: Omitted.

theorem
The Wijsman rλ α -statistical convergence to the functions of limit set of a triple sequence to the functions of (f mnk ) is closed.

Proof
If , then it is true.Assume that then we can choose a triple sequence of functions of such that as m,n,k→∞.If we prove that , then the proof will be complete.