Bernstein operator of rough I-core of triple sequences

We introduce and study some basic properties of Bernstein-Stancu polynomials of rough I-convergent of triple sequence spaces and also study the set of all Bernstein-Stancu polynomials of rough I-limits of a triple sequence spaces and relation between analytic ness and Bernstein-Stancu polynomials of rough I-core of a triple sequence spaces.


Introduction
The notion of rough convergence has firstly been presented by Phu [10][11][12] in finite dimensional normed spaces. The author also illustrated that this set LIM r x is closed, convex and bounded; and he at the same time proposed the concept about the rough Cauchy sequence. The author again examined the connections among rough convergence and the other types of convergence and the dependence for LIM r x on the roughness having degree of r. Aytar [1] investigated rough statistical convergence and described rough statistical limit points set for a given sequence and had two important criteria about statistical convergence related to the set and proved the fact that the set is convex and closed. Meanwhile, Aytar [2] investigated that the r-limit set of the sequence equals to the intersection of those sets and that r-core of the sequence equals to the union of those sets. Dundar and Cakan [9] investigated rough ideal convergence and defined the set of rough ideal limit points of a sequence. The concept of I-convergence of a triple sequence spaces depended on the very structure for the ideal I of the subsets of N 3 , where N denotes the natural numbers, is an expected result of the concept of convergence and statistical convergence.
In the present manuscript, we investigate several fundamental characteristics for rough I-convergence of a triple sequence spaces in three dimensional matrix spaces. We analyze the set of all rough I-limits consisting of a triple sequence spaces and at the same time the connection among analytic ness and rough I-core of a triple sequence spaces.
Let us assume that K is a subset consisting of the set consisting of positive integer numbers N 3 and let us assume this set K ik = {(m, n, k) ∈ K : m ≤ i, n ≤ j, k ≤ }. Under these assumptions, the natural density for K can be denoted by δ (K) = lim i, j, →∞ |Kij | i j , where K i j stands for the number of elements in K i j .
First applied the notion of (p, q)-calculus in approximation theory and put forward the (p, q)-analogue of Bernstein operators. Later, depending on (p, q)-integers, several approximation results for Bernstein-Stancu, Bernstein-Kantorovich, (p,q)-Lorentz, Bernstein-Schurer, Bleimann, Butzer and Hahn operators etc. In recent years, Khalid et al. have presented a very nice application using computer-aided geometric design and also applied those Bernstein basis to construct (p, q)-Bezier curves and surfaces depending upon (p, q)-integers, being a step other generalization of q-Bezier surfaces and curves.
Motivated by the above mentioned work on (p, q)-approximation and its application, in this paper we study statistical approximation properties of Bernstein-Stancu Operators depending on (p, q)-integers. Now we are going to remember several fundamental descriptions related to (p, q)-integers.
For any u, v, w ∈ N, the (p, q)-integer [uvw] p,q is described as We can describe the (p, q)-binomial coefficient as The equality about (p, q)-binomial expansion can be given as The Bernstein operator of order (r, s, t) is presented as follows x m+n+k · (r−m−1) x m+n+k · (r−m−1) Note that for η = µ = 0, (p, q)-Bernstein-Stancu operators in Eq. (1.2) reduces into (p, q)-Bernstein operators. Also for p = 1, (p, q)-Bernstein-Stancu operators in Eq. (1.1) result in q-Bernstein-Stancu operators. A triple sequence (either complex or real) may be described as a function x : N 3 → R (C), in which the complex, real and natural numbers are denoted by C, R and N, respectively. The various types of concepts for triple sequence has been first proposed and examined initially by Datta et al. [7], Debnath et al. [8], Esi et al. [3][4][5][6], Sahiner et al. [13,14], Subramanian et al. [15] and several others.
It is said that a triple sequence x = (x mnk ) is triple analytic when sup m,n,k Λ 3 usually denotes all triple analytic sequences space.

Definitions and Preliminaries
For the rest of the manuscript R 3 symbolizes the real three dimensional case having the metric. Take into consideration a triple sequence x = (x mnk ) such that x mnk ∈ R 3 ; m, n, k ∈ N 3 . The following definition are obtained:

Definition 2.4 It is assumed that f is a prescribed continuous function defined on
φ and r is known as a rough convergence degree of S rst,p,q ( f, x). If r = 0 then it is ordinary convergence of triple sequence of Bernstein-Stancu polynomials.
When this happens, ( f, x) is known as r-statistical limit of B mnk ( f, x). If r = 0 then it is ordinary statistical convergent of triple sequence of Bernstein polynomials.

Definition 2.6 It is said that a class I of subsets consisting of a nonempty set X is an ideal in
In this case (B mnk ( f, x)) is called rI-convergent to ( f, x) and a triple sequence of Bernstein polynomials (B mnk ( f, x)) is called rough I-convergent to ( f, x) with r as roughness of degree. If r = 0 then it is ordinary I-convergent.

Note 2.1 Generally, it is assumed that f is a continuous function on [0, 1].
A triple sequence of Bernstein polynomials (B mnk (g, x)) is not I-convergent in usual sense and |B mnk ( f, x) − B mnk (g, x)| ≤ r for all (m, n, k) ∈ N 3 or (m, n, k) ∈ N 3 : |B mnk ( f, x) − B mnk (g, x)| ≥ r ∈ I for some r > 0. Then the triple sequence of Bernstein polynomials (B mnk ( f, x)) is rI-convergent. Note 2.2 It is clear that rI-limit of a sequence B mnk ( f, x) of Bernstein polynomial is not necessarily unique.  ( f, x)) is a metric space (X, d) iff for each > 0 the set The set consisting of all I-accumulation points of (B mnk ( f, x)) is symbolized by I (Γ (B mnk ( f, x))).

Definition 2.11 Suppose that f is a continuous function on [0, 1]. It is said that a triple sequence of Bernstein polynomials (B mnk ( f, x)) is rough
φ for a triple sequence of Bernstein polynomials (B mnk ( f, x)) of real numbers, under these conditions we conclude Proof. Assume that diam (LIM r B mnk ( f, x)). Then, , x), we have A 1 ( ) ∈ I and A 2 ( ) ∈ I for every > 0, where

Main Results
Using the properties F (I), we get (A 1 ( ) c A 2 ( ) c ) ∈ F (I). Thus we write, and this is an clear contradiction. Hence diam (LIM r B mnk ( f, x)) ≤ 2r. Now, consider a triple sequence of Bernstein polynomials of (B mnk ( f, x)) of real numbers so that Let > 0. So, it can be written that (m, n, k) ∈ N 3 :   ( f, x)) of real numbers, I ⊂ 3 N is an admissible ideal . For a given arbitrary ( f, c) Proof. If we suppose, on the contrariwise, that there is a point ( f, c) ∈ I (Γ x ) and (( f, x)) .

Theorem 3.3 Assume that f is a continuous function on [0, 1]. A triple sequence of Bernstein polynomials
Proof. Necessity: By Theorem 3.1. Sufficiency: . Therefore the triple sequence spaces of Bernstein polynomials of (B mnk ( f, x)) is I-analytic. Let us suppose that ( f, x) has got another I-cluster point ( f , x) which is differing from ( f, x). The point x). It is not possible as f , x − ( f, x) = r and I − LIM r B mnk ( f, x) =B r (( f, x)). Because ( f, x) is the unique I- Proof. We know that I − LIM r B mnk ( f, x) φ, a triple sequence of Bernstein polynomials of (B mnk ( f, x)) is I-analytic. The number I − lim inf B mnk ( f,x) is an I-cluster point of ( f, x) and consequently, we have , x). Likewise, it can be seen that I − lim sup x mnk ∈ I − LIM 2r x mnk .
And it can easily be seen that

Proof.
When the given set I − core {B mnk ( f, x)} is singleton, then radius (I − core {B mnk ( f, x)}) = 0 and the triple sequence of Bernstein polynomials is I-convergent, i.e., Therefore we obtain Let us take the assumption that the set I − core {B mnk ( f, x)} is now not a singleton. It can now be written as I   ( f, x)) of real numbers, then Example 3.1 Using the symbolic programming software Matlab, we have shown the comparisons and some easy-to-understand graphics for the convergence of operators given by (1.2) to the function f (x) = 1 + x 3 sin(14x) for various parameters. Looking at figure 1(a), it can be seen that as the q and p approaches towards 1 provided 0 < q < p ≤ 1, (p, q)-Bernstein-Stancu operators presented by (1.2) converges towards the function f (x) = 1 + x 3 sin(14x). From figure 1(a) and (b), it can be observed that for η = µ = 0, as the value the (r, s, t) increases, (p, q)-Bernstein-Stancu operators (1.2) converges towards the function. Similarly from figure 2(a), it can be noticed that for η = µ = 5, as the value the q and p approaches towards 1 or some thing else provided 0 < q < p ≤ 1, (p,q)-Bernstein-Stancu operators by (1.2) converges towards the function. From figure 2(a) and (b), it is seen that when the value the [r,s,t] increases, (p, q)-Bernstein-Stancu operators which are presented by f(x) = 1+x 3 sin(14x) converges towards the function.