Triple rough statistical convergence of sequence of Bernstein operators

Article history: Received 24 November 2016 Received in revised form 26 January 2017 Accepted 26 January 2017 In this paper, using the concept of natural density, we introduce the notion of Bernstein operator of rough statistical convergence of triple sequence. We define the set of Bernstein operator of rough statistical limit points of a triple sequence spaces and obtain Bernstein operator of statistical convergence criteria associated with this set. Later, we prove that this set is closed and convex and also examine the relations between the set of Bernstein operator of rough statistical cluster points and the set of Bernstein operator of rough statistical limit points of a triple sequences.


Introduction
*The idea of statistical convergence was introduced by Steinhaus (1951) and also independently by Fast (1951) for real or complex sequences. Statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence.
The Bernstein operator of order ( , , ) is given by: In this case, we write − ( , ) = ( ) or ( , ) → ( ). If a triple sequence is statistically convergent, then for every > 0, infinitely many terms of the sequence may remain outside the − 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 satisfies some property P for all m, n, k except a set of natural density zero, then we say that the triple sequence satisfies P for almost all (m, n, k) and we abbreviate this by a.a. (m, n, k).
Let (x m i n j k ℓ ) be a subsequence 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 m i n j k ℓ ) is called a non thin subsequence 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 .
Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B rst (f, x)) 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 (2001), 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 (2008) 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. (2013) extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence.
In this paper, we introduce the notion of Bernstein operator of rough statistical convergence of triple sequences. Defining the set of Bernstein polynomials of rough statistical limit points of a triple sequence, we obtain to Bernstein operator of statistical convergence criteria associated with this set. Later, we prove that this set of Bernstein operator of statistical cluster points and the set of rough statistical limit points of a triple sequence.
Throughout the paper let r is a nonnegative real number.

Definitions and preliminaries
Definition 2.1: Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B mnk (f, x)) is said to be r − convergent to f(x) denoted by B mnk (f, x) → r f(x), provided that, The set has natural density zero for every ε > 0, or equivalently, if the condition, In addition, we can write B mnk (f, x) → rst f(x) if and only if the inequality, holds for every ε > 0 and almost all (m, n, k). Here r is called the Bernstein operator of roughness of degree. If we take r = 0, then we obtain the ordinary statistical convergence of triple sequence.
In a similar fashion to the idea of classic rough convergence, the idea of Bernstein operator of rough statistical convergence of a triple sequence spaces can be interpreted as follows: Assume that f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B mnk (f, y)) is statistically convergent and cannot be measured or calculated exactly; one has to do with an approximated (or statistically approximated) triple sequence of Bernstein polynomials (B mnk (f, x)) satisfying |B mnk (f, x) − B mnk (f, y)| ≤ r for all m, n, k (or for almost all (m, n, k), i.e., δ({(m, n, k) ∈ ℕ 3 : |B mnk (f, x) − B mnk (f, y)| > r}) = 0.
In general, the rough statistical limit of a triple sequence of Bernstein polynomials (B mnk (f, x)) may not unique for the roughness degree r > 0. So we have to consider the so called r − statistical limit set of a triple sequence of Bernstein polynomials (B mnk (f, x)), which is defined by: The triple sequence of Bernstein polynomials for a triple sequence of Bernstein polynomials (B mnk (f, x)) of real numbers, then we have: We know that LIM r = ϕ for an unbounded triple sequence of Bernstein polynomials (B mnk (f, x)). But such a triple sequence of Bernstein polynomials might be rough statistically convergent. For instance, define: It can be seen from the above example fact that the fact st − LIM r (B mnk (f, x)) ≠ ϕ does not imply LIM r (B mnk (f, x)) ≠ ϕ. Because a finite set of natural numbers has natural density zero, LIM r (B mnk (f, x)) ≠ ϕ implies st − LIM r (B mnk (f, x)) ≠ ϕ.
Because diam (B ̅ r (f(x))) = 2r, this shows that in general, the upper bound 2r of the diameter of the set st − LIM r (B rst (f, x)) is not a lower bound. Theorem 3.2: Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B mnk (f, x)) is r − statistically convergent to f(x) if and only if there exists a triple sequence of Bernstein polynomials (B rst (f, x)) such that st − lim (B mnk (f, y)) = f(x) and |B mnk (f, x) − B mnk (f, y)| ≤ r for each (m, n, k) ∈ ℕ 3 . Proof. Necessity: Assume that B mnk (f, x) → rst f(x). Then we have then, we write We have for all m, n, k ∈ ℕ 3 . By equation (  Then the set st − LIM r ′ (B mnk (f, x)) contains the origin of ℝ. So we have st − LIM r (B mnk (f, x)) ≠ ϕ.
Then we say that almost all triple sequence are contained in some ball with any radius greater than r. So the triple sequence space of Bernstein polynomials (B mnk (f, x)) is statistically analytic. (B mnk (f, y)) ⊆ st − LIM r (B mnk (f, x)) such that (B mnk (f, y)) → r f(x) as m, n, k → ∞. If we prove that f(x) ∈ st − LIM r (B mnk (f, x)), then the proof will be complete.

Conclusion
In recent years the statistical convergence has been adapted to the sequences of fuzzy numbers, interval numbers etc. In this paper we have studied the notion of Bernstein operator of rough statistical convergence of triple sequence. We define the set of Bernstein operator of rough statistical limit points of a triple sequence spaces and obtain Bernstein operator of statistical convergence criteria associated with this set. We proved some theorems of the introduced sequence spaces. This notion can be used for further generalization of such sequence spaces.