On triple sequence space of Bernstein operator of rough I λ - statistical convergence of weighted g ( A )

Abstract

We introduce and study some basic properties of rough $I_{λ}$ – statistical convergent of weight g (A), where g : $N^{3} \to [0, \infty)$ is a function statisying g (m, n, k) → ∞ and g (m, n, k) ↛ 0 as m, n, k → 0 of triple sequence of Bernstein polynomials, where A represent the RH-regular matrix and also prove the Korovkin approximation theorem by using the notion of weighted A-statistical convergence of weight g (A) limits of a triple sequence of Bernstein polynomials.

Keywords

Triple sequences rough convergence closed and convex cluster points and rough limit points Bernstein polynomials

1 Introduction

The idea of rough convergence was first introduced by Phu [10-12] in finite dimensional normed spaces, showed that the set ${LIM}_{x}^{r}$ is bounded, closed, convex and introduced the notion of rough Cauchy sequence also investigated the relations between rough convergence and other convergence types and the dependence of ${LIM}_{x}^{r}$ on the roughness of degree r.

Aytar [1] studied of rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained to statistical convergence criteria associated with this set and prove that this set is closed and convex. Also, Aytar [2] studied that the r-limit set of the sequence is equal to intersection of these sets and that r-core of the sequence is equal to the union of these sets. Dündar and Cakan [9] investigated of rough ideal convergence and defined the set of rough ideal limit points of a sequence. The notion of I-convergence of a triple sequence which is based on the structure of the ideal I of subsets of $ℕ \times ℕ \times ℕ$ , where $ℕ$ is the set of all natural numbers, is a natural generalization of the notion of convergence and statistical convergence and also Zhan et al. [16-21] studied various rough sets.

Our primary interest in the present paper is to obtain a general Korovkin-type approximation theorem for triple sequences of positive linear operators of two variables from H _w (K) to C _w (K) via statistical A-summability.

Let A be a three dimensional summability matrix. For a given triple sequence x = (x _mnk), the A-transform of x, denoted by Ax : = ((Ax) _ijℓ), given by

${(Ax)}_{i, j, ℓ} = \sum_{(m, n, k) \in ℕ^{3}} a_{i, j, ℓ, m, n, k} x_{mnk}$ (1.1) provided the triple series converges in Pringsheim sense for every $(i, j, ℓ) \in ℕ^{3}$ .

A three dimensional matrix A = (a _{i,j,ℓ,m,n,k}) is said to be RH-regular it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit. A three dimensional matrix A = (a _{i,j,ℓ,m,n,k}) is RH-regular if and only if

$P - lim_{i, j} a_{i, j, ℓ, m, n, k} = 0$ for each $(m, n, k) \in ℕ^{3}$ ,

$P - lim_{i, j, ℓ} \sum_{(m, n, k) \in ℕ^{3}} a_{i, j, ℓ, m, n, k} = 1$ ,

$P - lim_{i, j, ℓ} \sum_{m \in ℕ} a_{i, j, ℓ, m, n, k} = 0$ for each $n, k \in ℕ$ ,

$P - lim_{i, j, ℓ} \sum_{n \in ℕ} a_{i, j, ℓ, m, n, k} = 0$ for each $m, k \in ℕ$ ,

$P - lim_{i, j, ℓ} \sum_{k \in ℕ} a_{i, j, ℓ, m, n, k} = 0$ for each $m, n \in ℕ$ ,

$\sum_{(m, n, k) \in ℕ^{3}} | a_{i, j, ℓ, m, n, k} |$ is P-convergent for every $(i, j, ℓ) \in ℕ^{3}$ ,

there exist finite positive integers A and B such that ∑_m,n,k>B|a _{i,j,ℓ,m,n,k}| < A holds for every $(i, j, ℓ) \in ℕ^{3}$ .

Now let A = (a _{i,j,ℓ,m,n,k}) be a non-negative RH-regular summability matrix, and $K \subset ℕ^{3}$ . Then the A-density of K is given by $δ_{2}^{A} {K} : = P - lim_{i, j, ℓ} \sum_{(m, n, k) \in K (\in)} a_{i, j, ℓ, m, n, k}$ where $K (\in) : = {(m, n, k) \in ℕ^{3} : | x_{mnk} - L | \geq \in}$ provided that the limit on the right-hand side exists in Pringsheim sense. A real triple sequence x = (x _mnk) is said to be A-statistically convergent to a number L if, for every ∈ > 0, $δ_{2}^{A} {(m, n, k) \in ℕ^{3} : | x_{mnk} - L | \geq \in} = 0 .$

In this case, we write ${st}_{2}^{A} - lim_{m, n, k} = L$ .

Let K be a subset of the set of positive integers $ℕ \times ℕ \times ℕ$ and let us denote the set K _ikℓ ={ (m, n, k) ∈ K : m ≥ i, n ≤ j, k ≤ ℓ }. Then the natural density of K is given by $δ (K) = lim_{i, j, ℓ \to \infty} \frac{| K_{ij ℓ} |}{ij ℓ},$ where |K _ijℓ| denotes the number of elements in K _ijℓ.

The Bernstein operator of order (r, s, t) is given by $B_{r s t} (f, x) = \sum_{m = 0}^{r} \sum_{n = 0}^{s} \sum_{k = 0}^{t} f (\frac{m n k}{r s t}) (\begin{array}{l} r \\ m \end{array}) (\begin{array}{l} s \\ n \end{array}) \cdot (\begin{array}{l} t \\ k \end{array}) x^{m + n + k} {(1 - x)}^{(m - r) + (n - s) + (k - t)}$ where f is a continuous (real or complex valued) function defined on [0, 1].

Throughout the paper, $ℝ$ denotes the real of three dimensional space with metric (X, d). Consider a triple sequence of Bernstein polynomials (B _mnk (f, x)) such that $(B_{mnk} (f, x)) \in ℝ$ , $m, n, k \in ℕ^{3}$ .

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 convergent to $0 \in ℝ$ , written as st - limx = 0, provided that the set $K_{\in} : = {(m, n, k) \in ℕ^{3} : | B_{mnk} (f, x) - f (x) | \geq \in}$ has natural density zero for any ∈ > 0. In this case, 0 is called the statistical limit of the triple sequence of Bernstein polynomials. i.e., δ(K_∈) = 0. That is $\lim_{r s t \to \infty} \frac{1}{p q j} | {(m, n, k) \leq (p, q, j) : | B_{m n k} (f, x) - (f, x) | \geq \in} | = 0.$

In this case, we write δ - lim B _mnk (f, x) = f (x) or B _mnk (f, x) → ^{S
_B}f (x). A subset A of $ℕ^{3}$ is said to have asymptotic density d (A) if $d (A) = lim_{ij ℓ \to \infty} \frac{1}{ij ℓ} \sum_{m = 1}^{i} \sum_{n = 1}^{j} \sum_{k = 1}^{ℓ} χ_{A} (K) .$

A triple sequence (real or complex) can be defined as a function $x : ℕ \times ℕ \times ℕ \to ℝ (ℂ)$ , 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. [13, 14], Esi et al. [3-6], Datta et al. [7], Subramanian et al. [15], Debnath et al. [8] and many others.

A triple sequence of Bernstein polynomials is said to be triple Bernstein polynomials of analytic if $sup_{m, n, k} {| B_{mnk} (f, x) - f (x) |}^{\frac{1}{m + n + k}} < \infty .$

The space of all triple of Bernstein polynomials of analytic sequences are usually denoted by $Λ_{B}^{3}$ .

2 Definitions and Preliminaries

Throughout the paper $ℝ^{3}$ denotes the real three dimensional case with the metric. Consider a triple sequence x = (x _mnk) such that $x_{mnk} \in ℝ^{3}$ ; $m, n, k \in ℕ \times ℕ \times ℕ$ . The following definition are obtained:

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 statistically convergent to f (x) denoted by B _mnk (f, x) → ^st-limxf (x), if for any ∈ > 0 we have d (A (∈)) = 0, where $A (\in) = {(m, n, k) \in ℕ^{3} : | B_{mnk} (f, x) - f (x) | \geq \in} .$

Definition 2.2. Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) is said to be statistically convergent to f (x) denoted by B _mnk (f, x) → ^st-limxf (x), provided that the set ${(m, n, k) \in ℕ^{3} : | B_{mnk} (f, x) - f (x) | \geq \in},$ has natural density zero for every ∈ > 0.

In this case, f (x) is called the statistical limit of the sequence of Berstein polynomials.

Definition 2.3. Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) in a metric space (X, | . ,. |) and r be a non-negative real number is said to be r-convergent to f (x), denoted by B _mnk (f, x) → ^rf (x), if for any ∈ > 0 there exists $N_{\in} \in ℕ^{3}$ such that for all m, n, k ≥ N _∈ we have $| B_{mnk} (f, x) - f (x) | < r + \in$ In this case B _mnk (f, x) is called an r-limit of f (x).

Remark 2.4. We consider r-limit set B _mnk (f, x) which is denoted by ${LIM}_{B_{mnk} (f, x)}^{r}$ and is defined by ${LIM}_{B_{m n k} (f, x)}^{r} = {B_{mnk} (f, x) \in X : B_{mnk} (f, x) \to^{r} f (x)} .$

Definition 2.5. 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 if ${LIM}_{B_{mnk} (f, x)}^{r} \neq φ$ and r is called a rough convergence degree of B _mnk (f, x). If r = 0 then it is ordinary convergence of triple sequence of Bernstein polynomials.

Definition 2.6. Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) in a metric space (X, | . ,. |) and r be a non-negative real number is said to be r-statistically convergent to f (x), denoted by B _mnkf (x) → ^r-st₃f (x), if for any ∈ > 0 we have d (A (∈)) = 0, where $A (\in) = {(m, n, k) \in ℕ^{3} : | B_{m nk} f (x) - f (x) | \geq r + \in} .$

In this case f (x) is called r-statistical limit of B _mnkf (x). If r = 0 then it is ordinary statistical convergent of triple sequence of Bernstein polynomials.

Definition 2.7. A class I of subsets of a nonempty set X is said to be an ideal in X provided

$φ \in I .$

A, B ∈ I implies A ⋃ B ∈ I.

A ∈ I, B ⊂ A implies B ∈ I.

I is called a nontrivial ideal if X ∉ I.

Definition 2.8. A nonempty class F of subsets of a nonempty set X is said to be a filter in X. Provided

φ ∈ F.

A, B ∈ F implies A ⋂ B ∈ F.

A ∈ F, A ⊂ B implies B ∈ F.

Definition 2.9. I is a non trivial ideal in X,X ≠ φ, then the class $F (I) = {M \subset X : M = X \ A for some A \in I}$ is a filter on X, called the filter associated with I.

Definition 2.10. A non trivial ideal I in X is called admissible if {x } ∈ I for each x ∈ X.

Note 2.11. If I is an admissible ideal, then usual convergence in X implies I convergence in X.

Remark 2.12. If I is an admissible ideal, then usual rough convergence implies rough I-convergence.

Definition 2.13. Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) in a metric space (X, | . ,. |) and r be a non-negative real number is said to be rough ideal convergent of weight g or rI _λ-convergent to f (x), denoted by $B_{mnk} \to^{{rI}_{λ}^{g}} f (x)$ , if for any ∈ > 0 we have ${(p, q, j) \in ℕ^{3} : \frac{1}{g (λ_{pqj})} | B_{mnk} (f, x) - f (x) | \geq r + \in} \in I .$ In this case (B _mnk (f, x)) is called rI _λ-limit of (f, x) and a triple sequence of Bernstein polynomials (B _mnk (f, x)) is called rough I _λ- convergent weight g to f (x) with r as roughness of degree. If r = 0 then it is ordinary I _λ-convergent of weight g.

Note 2.14. Generally, Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, y)) is not I _λ-convergent of weight g in usual sense and |B _mnk (f, x) - B _mnk (f, y) | ≤ r for all $(m, n, k) \in ℕ^{3}$ or ${(p, q, j) \in ℕ^{3} : \frac{1}{g (λ_{pqj})} | B_{mnk} (f, x) - B_{mnk} (f, y) | \geq r} \in I .$ for some r > 0. Then the triple sequence of Bernstein polynomials (B _mnk (f, x)) is rI _λ- convergent of weight g.

Note 2.15. It is clear that ${rI}_{λ}^{g} -$ limit of f (x) is not necessarily unique.

Definition 2.16. Consider ${rI}_{λ}^{g} -$ limit set of f (x), which is denoted by $I_{λ}^{g} - {LIM}_{B_{m nk} (f, x)}^{r} = {f (x) \in X : B_{mnk} (f, x) \to^{{rI}_{λ}^{g}} f (x)},$ then the triple sequence of Bernstein polynomials (B _mnk (f, x)) is said to be rI _λ-convergent of weight g, if $I_{λ}^{g} - {LIM}_{B_{mnk} (f, x)}^{r} \neq φ$ and r is called a rough I _λ-convergence of weight g degree of B _mnk (f, x).

Definition 2.17. 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 $I_{λ}^{g} -$ analytic if there exists a positive real number M such that ${(p, q, j) \in ℕ^{3} : \frac{1}{g (λ_{pqj})} {| B_{mnk} (f, x) |}^{1 / m + n + k} \geq M} \in I .$

Definition 2.18. A point f (x) ∈ X is said to be an $I_{λ}^{g} -$ accumulation point and Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) in a metric space (X, d) if and only if for each ∈ > 0 the set ${(p, q, j) \in ℕ^{3} : d (B_{m n k} (f, x), f (x)) = \frac{1}{g (λ_{p q j})} | B_{m n k} (f, x) - f (x) | < \in} \notin I .$

We denote the set of all $I_{λ}^{g} -$ accumulation points of (B _mnk (f, x)) by $I_{λ}^{g} (Γ_{B_{mnk} (f, x)})$ .

Definition 2.19. Let f be a continuous function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B _mnk (f, x)) of real numbers, the notions of ideal limit superior and ideal limit inferior are defined as follows: $I_{λ}^{g} - \lim \sup B_{m n k} (f, x) = {\begin{array}{l} \sup B_{x}, if B_{B_{m n k} (f, x)} \neq φ \\ - \infty, if B_{B_{m n k} (f, x)} = φ \end{array}$ and $I_{λ}^{g} - \lim \inf (B_{m n k} (f, x)) = {\begin{array}{l} \inf A_{B_{m n k} (f, x)}, if A_{B_{m n k} (f, x)} \neq φ \\ + \infty, if A_{B_{m n k} (f, x)} = φ \end{array}$ where $A_{B_{m n k} (f, x)} = {a \in ℝ : {(m, n, k) \in ℕ^{3} : B_{m n k} (f, x) < a} \notin I}$ and $B_{B_{m n k} (f, x)} = {b \in ℝ : {(m, n, k) \in ℕ^{3} : B_{m n k} (f, x) < b} \notin I} .$

Definition 2.20. 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 rough I _λ-convergent of weight g, if $I_{λ}^{g} - {LIM}^{r} B_{mnk} (f, x) \neq φ$ . It is clear that if $I_{λ}^{g} - {LIM}^{r} B_{mnk} (f, x) \neq φ$ for a triple sequence of Bernstein polynomials (B _mnk (f, x)) of real numbers, then we have $I_{λ}^{g} - L I M^{r} B_{m n k} (f, x) = [I_{λ}^{g} - \lim \sup B_{m n k} (f, x) - r, I_{λ}^{g} - \lim \inf B_{m n k} (f, x) + r] .$

Remark 2.21. Let $λ = {(λ_{pqj})}_{pqj \in ℕ^{3}}$ be a non-decreasing sequence of positive numbers tending to ∞ such that $λ_{(pqj) + 1} \leq λ_{pqj} + 1, λ_{111} = 1 .$

The collection of such sequences λ will be denoted by η.

We define the generalized de la Vallée-Pousin mean of weight g by $t_{pqj} (x) = \frac{1}{g (λ_{pqj})} \sum_{(m, n, k) \in I_{pqj}} x_{mnk}$ where I _rst = [(pqj) - λ _pqj+1, pqj].

Let r = (r _mnk) be a triple sequence of nonnegative numbers such that r ₀₀₀ > 0 and

$u_{pqj} = \sum_{m, n, k}^{p, q, j} u_{mnk} \to \infty .$ (2.1) $t_{p q j} (x) = \frac{1}{g (λ_{p q j})} \frac{1}{u_{p g j}} \sum_{(m, n, k) \in I_{p q j}} u_{p q j} x_{m n k}, p, q, j = 0, 1, 2, \dots$ where $I_{r s t} = [(p q j) - λ_{p q j + 1}, p q j]$ .

Definition 2.22. 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 [V, λ] (I) ^g-summable to f (x), if $I - lim_{pqj} t_{pqj} (B_{mnk} (f, x)) \to f (x) .$ i.e., for any δ > 0, ${(pqj) \in ℕ^{3} : | t_{pqj} (B_{mnk} (f, x)) - f (x) | \geq δ} \in I$ and it is denoted by [V, λ] (I) ^g.

Definition 2.23. 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 I _λ-statistically convergent of weight g, if for every $\begin{array}{l} \in > 0 and δ >0 \\ {(p, q, j) \in ℕ^{3} : \frac{1}{g (λ_{p q j})} | {(m, n, k) \in I_{p q j} : | B_{m n k} (f, k) - f (x) | \geq r + \in} | \geq δ} \in I . \end{array}$

In this case we write (I _λ) ^g - lim B _mnk (f, x) = f (x). Or B _mnk (f, x) → f (x) (I _λ) ^g.

Definition 2.24. 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 I _λ-statistically convergent of weight g, if for every ∈ > 0 $\lim_{p q j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} | {(m, n, k) \in I_{p q j} : | B_{m n k} (f, k) - f (x) | \geq r + \in} | \in I .$

In this case we write ${(I_{λ})}^{g_{\bar{N}}} - lim B_{mnk} (f, x) = f (x)$ . Or $B_{mnk} (f, x) \to f (x) {(I_{λ})}^{g_{\bar{N}}}$ .

Remark 2.25. If u _pqj = 1 for all m, n, k, then $(\bar{N}, u_{pqj}) -$ summable is reduced to [C , 1, 1, 1] -summable or Cesáro summable and statistically convergent of weight g is reduced to statistical convergence of Bernstein polynomials of triple sequence.

3 A Korovkin-type approximation theorem by g(A) -statistical convergence

Let C _B (K) the space of all continuous and bounded real valued functions on K = [0, ∞) × [0, ∞) × [0, ∞). This space is equipped with the supremum norm $| | f | | = sup_{(x, y, z) \in K} B_{mnk} | f, (x, y, z) |, (f \in C_{B} (K)) .$

Consider the triple space of H _w (K) of all real valued functions of Bernstein polynomials of f on K satisfying $| B_{m n k} (f, (u, v, w)) - B_{m n k} (f, (x, y, z)) | \leq w (| \frac{u}{1 + u} - \frac{x}{1 + x} |, | \frac{v}{1 + v} - \frac{y}{1 + y} |, | \frac{w}{1 + w} - \frac{z}{1 + z} |)$ where w be a function of the type of the modulus of continuity given by, for δ, δ ₁, δ ₂, δ ₃ > 0,

w is non-negative increasing function on K with respect to δ ₁, δ ₂, δ ₃,

w (δ, δ ₁ + δ ₂ + δ ₃) ≤ w (δ, δ ₁) + w (δ, δ ₂) + w (δ, δ ₃),

w (δ ₁ + δ ₂ + δ ₃, δ) ≤ w (δ ₁, δ) + w (δ ₂, δ) + w (δ ₃, δ),

$lim_{δ_{1}, δ_{2} δ_{3} \to 0} w (δ_{1}, δ_{2}, δ_{3}) = 0$ .

The Bernstein polynomials of B _mnk (f) ∈ H _w (K) satisfies the inequality $B_{m n k} | (f, (x, y, z)) | \leq B_{m n k} (f, (0, 0, 0)) + w (1, 1, 1), x, y, z \geq 0$ and hence it is bounded on K. Therefore H _w (K) ⊂ C _B (K).

We also use the following Bernstein polynomials of test functions $\begin{matrix} B_{mnk} (f_{000}, (u, v, w)) & = 1, B_{mnk} (f_{111}, (u, v, w)) = \frac{u}{1 + u}, \\ B_{mnk} (f_{222}, (u, v, w)) & = \frac{v}{1 + v}, B_{mnk} (f_{333}, (u, v, w)) = \frac{w}{1 + w} \end{matrix}$ and $B_{mnk} (f_{444}, (u, v, w)) = {(\frac{u}{1 + u})}^{2} + {(\frac{v}{1 + v})}^{2} + {(\frac{w}{1 + w})}^{2} .$

Definition 3.1. Let f be a continuous function defined on the closed interval [0, 1] and A = (a _{i,j,ℓ,m,n,k}) be a nonnegative RH-regular matrix. A triple sequence of Bernstein polynomials (B _mnk (f, x)) is said to be I _λ-statistically convergent of weight g, if for every ∈ > 0

$lim_{pqj} \sum_{m, n, k \in E (u, \in)} a_{i, j, ℓ, m, n, k} = 0,$ (3.1) $E (u, \in) = {(p, q, j) \in ℕ^{3} : \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} | {(m, n, k) \in I_{p q j} : | B_{m n k} (f, x) - f (x) | \geq r + \in} |} \in I .$ In this case we write ${(I_{λ})}_{A}^{g_{\bar{N}}} - lim B_{mnk} (f, x) = f (x)$ . Or $B_{mnk} (f, x) \to f (x) {(I_{λ})}_{A}^{g_{\bar{N}}}$ .

4 Main results

Theorem 4.1. Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials of real numbers of (B_mnk (f,x)): [0,1] → [0,1] ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (f, x) - f (x) | = 0$ (4.1) if and only if ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (1, x) - 1 | = 0$ (4.2) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v, x) - x | = 0$ (4.3) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{2}, x) - x^{2} | = 0$ (4.4) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{3}, x) - x^{3} | = 0$ (4.5) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{4}, x) - x^{4} | = 0$ (4.6) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{5}, x) - x^{5} | = 0$ (4.7) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{6}, x) - x^{6} | = 0$ (4.8) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{7}, x) - x^{7} | = 0$ (4.9) ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (v^{8}, x) - x^{8} | = 0$ (4.10)

Proof. Following the proof of Theorem, we obtains $\begin{matrix} E \subset E_{1} \subset E_{2} \subset E_{3} & \subset E_{4} \subset E_{5} \\ \subset E_{6} \subset E_{7} \subset E_{8} \subset E_{9} \end{matrix}$ (4.11) and so $\begin{matrix} δ {(I_{λ})}^{g_{\bar{N}}} (E) & \subset δ {(I_{λ})}^{g_{\bar{N}}} (E_{1}) + δ {(I_{λ})}^{g_{\bar{N}}} (E_{2}) \\ + δ {(I_{λ})}^{g_{\bar{N}}} (E_{3}) + δ {(I_{λ})}^{g_{\bar{N}}} (E_{4}) \\ + δ {(I_{λ})}^{g_{\bar{N}}} (E_{5}) + δ {(I_{λ})}^{g_{\bar{N}}} (E_{6}) \\ + δ {(I_{λ})}^{g_{\bar{N}}} (E_{7}) + δ {(I_{λ})}^{g_{\bar{N}}} (E_{8}) \\ + δ {(I_{λ})}^{g_{\bar{N}}} (E_{9}) . \end{matrix}$ Equations (4.2)-(4.10) give that ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (f, x) - f (x) | = 0 . □$

Theorem 4.2. Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials of real numbers of line break (B_mnk (f,x)): [0,1] → [0,1] which satisfies (ref4.3) to (4.10) ofTheorem4.1and the following condition holds : ${(I_{λ})}^{g_{\bar{N}}} - \lim | B_{m n k} (1, x) - 1 | = 0$ (4.12)Then $\lim_{p q j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} \sum_{m, n, k = 0}^{p, q, j} u_{m n k} {| B_{m n k} (f, x) - f (x) | \geq r + \in} \in I .$ (4.13)

Proof. It follows (4.12) that |B _mnk (1, x) |^≤C′, for some constant C' > 0 and for all $m, n, k \in ℕ^{3}$ . Hence, for f ∈ [0, 1] we obtain $\begin{matrix} u_{mnk} {| B_{mnk} (f, x) - f (x) |} \\ \leq u_{mnk} (| f (x) | | B_{mnk} (1, x) | + | f (x) |) \leq u_{m n k} C (C^{'} + 1) \end{matrix}$ (4.14) In right hand side of (4.14) is constant. Hence we get ${(I_{λ})}^{g_{\bar{N}}} - lim | B_{mnk} (f, x) - f (x) | = 0$ .□

5 Statistical

A-summability

In this section we define statistical A-summability of a triple sequence of Bernstein polynomials of RH-regular summability matrix and prove that it is stronger than A-Bernstein polynomials of rough statistical convergence for analytic triple sequences.

Definition 5.1. Let f be a continuous function defined on the closed interval [0, 1] and A = (a _{i,j,ℓ,m,n,k}) be a nonnegative regular summability matrix of a triple sequence of Bernstein polynomials (B _mnk (f, x)) is said to be rough statistically A summable convergent to f (x) denoted by B _mnk (f, x) → ^st_A-limxf (x), if for every ∈ > 0 provided that the set $δ_{2} ({(m, n, k) \in ℕ^{3} : | B_{mnk} (f, x) - f (x) | \geq r + \in}) = 0,$

where B _mnk (f, x) is as in (1.1). So, if x is Bernstein polynomials of rough statistically A-summable to f (x), then for every ∈ > 0, $P - \lim_{p q j} \frac{1}{p q j} ({(m, n, k) \in ℕ^{3} : | B_{m n k} (f, x) - f (x) | \geq r + \in}) = 0.$

Thus, the triple sequence of Bernstein polynomials of B _mnk (f, x) is rough statistically A-summable to f (x) if and only if B _mnk (f, Ax) is rough statistically convergent to f (x).

Theorem 5.2. Let fbe a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers is analytic and A-rough statistically convergent to f(x) then it is rough statistically A-summable convergent to f(x) but not conversely.

Proof. Let B_mnk (f,x) be analytic and A-rough statistically convergent to f (x), and $K (\in) = {(m, n, k) \in ℕ^{3} : | B_{mnk} (f, x) - f (x) | \geq r + \in} .$ Then $\begin{array}{l} {| B_{m n k} (f, x) - f (x) |}^{1 / m + n + k} \\ = | \sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} {(B_{m n k} (f, x))}^{1 / m + n + k} - f (x) | \\ = | \sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} {(B_{m n k} (f, x) - f (x))}^{1 / m + n + k} + f (x) (\sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} - 1) | \\ \leq | \sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} {(B_{m n k} (f, x) - f (x))}^{1 / m + n + k} | + | f (x) | | \sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} - 1 | \\ \leq | \sum_{(m, n, k) = 1} a_{i, j, ℓ, m, n, k} {(B_{m n k} (f, x) - f (x))}^{1 / m + n + k} | + | \sum_{(m, n, k \notin K (\in))} a_{i, j, ℓ, m, n, k} {(B_{m n k} (f, x) - f (x))}^{1 / m + n + k} | + | f (x) | | \sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} - 1 | \end{array} .$ Using the definition of triple sequence of Bernstein polynomials of A-rough statistical convergence and the conditions of RH-regularity of A, we get P-lim_pqj| B_mnk (f,x) -f (x) | =0 from the arbitrariness of ∈ >0. Hence $B_{m n k} (f, x) \to^{A^{- \lim x}} f (x)$ . To show that the converse is not true in general, we give the following examples:

(i)A= (a_i,j,l,m,n,k) be C [1,1,1 , 1,1,1], the six dimensional Cesáro matrix $a_{i, j, ℓ, m, n, k} = {\begin{matrix} (1 / mnk), & if i \leq m, j \leq n, ℓ \leq k \\ 0, & otherwise \end{matrix}$ and let x= (x_mnk) be defined ${(B_{mnk} (f, x))}^{1 / m + n + k} = {(- 1)}^{mnk} for all m, n, k .$ Then triple sequence of Bernstein polynomials of B_mnk (f,x) is C [1,1,1 , 1,1,1] summable (and hence statistical C [1,1,1 , 1,1,1] -summable) to zero but not C [1,1,1 , 1,1,1] statistically convergent.

(ii) Define A= (a_i,j,l,m,n,k) by $a_{i, j, ℓ, m, n, k} = {\begin{array}{l} \frac{1}{m^{3} n^{3} k^{3}}, if m = n = k, i \leq m, j \leq n, ℓ \leq k and m, n, k are even \\ \frac{1}{(m^{3} - m) (n^{3} - n) (k^{3} - k)}, if m = n = k, i \neq j \neq k, i \leq m, j \leq n, ℓ \leq k and m, n, k are odd \\ 0, if otherwise \end{array}$ and define the triple analytic sequence of Bernstein polynomials of B_mnk (f,x) ${[B_{m n k} (f, k)]}^{1 / m + n + k} = {\begin{array}{l} 1, \\ if m, n, k are odd and for all m, n, k \\ 0, \\ otherwise \end{array}$ We can easily verify that A is RH-regular, that is, conditions RH(i)-RH (vi) hold. Moreover, for the sequence defined above, $\sum_{(m, n, k) = 1}^{\infty} a_{i, j, ℓ, m, n, k} {[B_{m n k} (f, x)]}^{1 / m + n + k} = {\begin{array}{l} 1 / 3 \\ if and m, n, k are even \\ (m + 1) (n + 1) (k + 1) / 27 m n k \\ if m, n, k are odd \\ 0 \\ if otherswise \end{array}$ Hence it is clear that the triple sequence of Bernstein polynomials of B_mnk (f,x) is not A-summable and hence is not A-rough statistically convergent but $B_{m n k} (f, k) \to^{A^{- \lim x}} f (x) = 0$ . Hence triple sequence of Bernstein polynomials of B_mnk (f,x) is rough statistically A-summable to zero.□

6 Korovkin-type approximation theorem

Theorem 6.1. Let f be a continuous function defined on the closed interval [0,1]. A triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers of C_B (K) into itself. Then for all f ∈ C_B (K), $P - lim_{pqj} | | B_{mnk} (f, x) - f (x) | |_{C_{B} (K)} = 0 .$ if and only if $P - \lim_{p q j} {‖ B_{m n k} (f_{u v w}, x) - f_{u v w} (x) ‖}_{C_{B} (K)} = 0, (u, v, w = 0, 1, 2, 3, \dots)$ where $\begin{matrix} B_{mnk} (f_{000}, (u, v, w)) & = 1, \\ B_{mnk} (f_{111}, (u, v, w)) & = \frac{u}{1 + u}, \\ B_{mnk} (f_{222}, (u, v, w)) & = \frac{v}{1 + v}, \\ B_{mnk} (f_{333}, (u, v, w)) & = \frac{w}{1 + w} \end{matrix}$ and $B_{mnk} (f_{444}, (u, v, w)) = {(\frac{u}{1 + u})}^{2} + {(\frac{v}{1 + v})}^{2} + {(\frac{w}{1 + w})}^{2} .$

Proof. It is routine verification. Therefore the proof is omitted.□

Theorem 6.2. Let f be a continuous function defined on the closed interval [0,1]. Let A= (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix and a triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers of C_B (K) into itself. Then for all f ∈ C_B (K), ${st}_{3 A} - lim_{pqj} | | B_{mnk} (f, x) - f (x) | |_{C_{B} (K)} = 0$ if and only if $s t_{3 A} - \lim_{p q j} {‖ B_{m n k} (f_{u v w}, x) - f_{u v w} (x) ‖}_{C_{B} (K)} = 0, (u, v, w = 0, 1, 2, 3, \dots)$ where $\begin{matrix} B_{mnk} (f_{000}, (u, v, w)) & = 1, \\ B_{mnk} (f_{111}, (u, v, w)) & = \frac{u}{1 + u}, \\ B_{mnk} (f_{222}, (u, v, w)) & = \frac{v}{1 + v}, \\ B_{mnk} (f_{333}, (u, v, w)) & = \frac{w}{1 + w} \end{matrix}$ and $B_{mnk} (f_{444}, (u, v, w)) = {(\frac{u}{1 + u})}^{2} + {(\frac{v}{1 + v})}^{2} + {(\frac{w}{1 + w})}^{2} .$

Proof. It is routine verification. Therefore the proof is omitted.□

Theorem 6.3. Let f be a continuous function defined on the closed interval [0,1]. Let A= (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix and a triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers of C_B (K) into itself. Then for all f ∈ C_B (K), ${st}_{3} - lim_{pqj} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f, x) - f (x) | |_{C_{B} (K)} = 0 .$ (6.1) if and only if ${st}_{3} - lim_{pqj} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{uvw}, x) - f_{uvw} (x) | |_{C_{B} (K)} = 0,$ (6.2) $(u, v, w = 0, 1, 2, 3, \dots)$ where $\begin{matrix} B_{mnk} (f_{000}, (u, v, w)) & = 1, \\ B_{mnk} (f_{111}, (u, v, w)) & = \frac{u}{1 + u}, \\ B_{mnk} (f_{222}, (u, v, w)) & = \frac{v}{1 + v}, \\ B_{mnk} (f_{333}, (u, v, w)) & = \frac{w}{1 + w} \end{matrix}$ and $B_{mnk} (f_{444}, (u, v, w)) = {(\frac{u}{1 + u})}^{2} + {(\frac{v}{1 + v})}^{2} + {(\frac{w}{1 + w})}^{2} .$

Proof. Condition (6.2) follows immediately from condition (6.1) since each f_uvw ∈ C_B (K) (u,v,w=0,1,2,3,…). Let us prove the converse. By the continuity of f on compact set K, we can write $| f (x, y, z) | \leq M$ , where $M = {‖ f ‖}_{C_{B} (K)}$ , for every ∈ >0, there is a number δ > 0 such that |f(u, v, w) – f(x, y, z)| < r + ∈ for all (u, v, w) ∈ K satisfying $| \frac{u}{1 + u} - \frac{x}{1 + x} | < δ$ ; $| \frac{v}{1 + v} - \frac{y}{1 + y} | < δ$ and $| \frac{w}{1 + w} - \frac{z}{1 + z} | < δ$ . Hence we get

$\begin{array}{l} | f (u, v, w) - f (x, y, z) | < r + ϵ + \frac{3 M}{δ^{2}} { {(\frac{u}{1 + u} - \frac{x}{1 + x})}^{2} \\ + {(\frac{v}{1+v} - \frac{y}{1+y})}^{2} + {(\frac{w}{%} 1+w - \frac{z}{1+z})}^{2}} \end{array}$ (6.3) From (6.3) we obtain for any m,n,k ∈ ℕ³ that $\begin{matrix} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f, x) - f (x) | | \leq \\ \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (| f (u, v, w) - f (x, y, z) |, (x, y, z)) + \\ | f (x, y, z) | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{000}, (x, y, z)) - f_{000} (x, yz) | \end{matrix}$

$\leq \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} B_{m n k} (r + ϵ + \frac{3 M}{%} δ^{2} { {(\frac{u}{1 + u} - \frac{x}{1 + x})}^{2} + {(\frac{v}{1 + v} - \frac{y}{1 + y})}^{2} + {(\frac{w}{1 + w} - \frac{z}{1 + z})}^{2} }, (x, y, z) ) + | f (x, y, z) | | \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot B_{mnk} (f_{000}, (x, y, z)) {-f}_{000} (x, yz) |$

$\begin{matrix} \leq \in + (\in + M) | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{000}, (x, y, z)) - f_{000} | \\ + \frac{3 M}{δ^{2}} | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{333}, (x, y, z)) - f_{333} (x, yz) | \end{matrix}$ $\begin{matrix} + 3 | x | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{111}, (x, y, z)) - f_{111} (x, yz) | \\ + 3 | y | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{222}, (x, y, z)) - f_{222} (x, y, z) | \\ + 3 | z | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{333}, (x, y, z)) - f_{333} (x, y, z) | \\ + ({(\frac{x}{1 + x})}^{2} + {(\frac{y}{1 + y})}^{2} + {(\frac{z}{1 + z})}^{2}) \cdot \\ | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{000}, (x, y, z)) - f_{000} (x, y, z) | \\ \leq r + \in + (r + \in + M + \frac{3 M}{δ^{2}} (C^{2} + D^{2} + E^{2})) \cdot \\ | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{000}, (x, y, z)) - f_{000} (x, y, z) | \\ + \frac{3 M}{δ^{2}} | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{333}, (x, y, z)) - f_{333} (x, y, z) | \\ + \frac{3 MC}{δ^{2}} | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{111}, (x, y, z)) - f_{111} (x, y, z) | \\ + \frac{3 MD}{δ^{2}} | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{222}, (x, y, z)) - f_{222} (x, y, z) | \\ + \frac{3 ME}{δ^{2}} | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{333}, (x, y, z)) - f_{333} (x, y, z) |, \end{matrix}$ where C:= max |x|, D:=max |y|, E:=max |z|. Taking supremum over (x,y,z) ∈ K we get $| | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f) - f | |$ $\begin{array}{l} \leq r + ϵ + B \sum_{(u, v, w) = 0}^{3} ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot \\ B_{m n k} (f_{u v w}, (x, y, z)) - f_{u v w} (x, y, z) ‖, \end{array}$ where $B : = \max {r + \in + M + \frac{3 M}{δ^{2}} (C^{2} + D^{2} + E^{2}), \frac{3 M}{δ^{2}}, \frac{6 M C}{δ^{2}}, \frac{6 M D}{δ^{2}}, \frac{6 M E}{δ^{2}}} .$ Now for given ρ > 0, choose ∈ >0 such that ∈ < ρ and define $\begin{array}{l} E : = {(m, n, k) \in ℕ^{3} : ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot \\ B_{mnk} (f, (x, y, z)) -f (x, y, z) ‖ \geq ρ} \\ E_{u v w} : = {(m, n, k) \in ℕ %^{3} : ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot \\ B_{m n k} (f_{u v w}, (x, y, z)) - f_{u v w} (x, y, z) ‖ \geq \frac{ρ - ϵ}{6 B}}, \\ u, v, w=0, 1, 2, 3 . \end{array}$ Then $E \subset \cup_{u, v, w = 0}^{3} E_{u v w}$ and so $δ_{2} (E) \leq \sum_{(u, v, w) = 0}^{3} δ_{2} (E_{uvw}) .$ By considering this inequality and using (6.2) we obtain (6.1).□

Example 6.4. Now, we will show that Theorem 6.3 is stronger than its classical and statistical forms. Let A be C [1,1,1 , 1,1,1] and defined x= (x_mnk) by x_mnk= (-1) ^mnk for all m,n,k. Then triple sequence of Bernstein polynomials is neither P-convergent nor A-rough statistically convergent but st₃–lim Ax = 0.

The Bernstein operator of order (u,v,w) is given by $B_{u v w} (f, (x, y, z)) = \sum_{m = 0}^{u} \sum_{n = 0}^{v} \sum_{k = 0}^{w} f (\frac{m n k}{u v w}) (\begin{array}{l} u \\ m \end{array}) \cdot (\begin{array}{l} v \\ n \end{array}) (\begin{array}{l} w \\ k \end{array}) x^{m + n + k} {(1 - x)}^{(m - u) + (n - v) + (k - w)}$ where f is a continuous (real or complex valued) function defined on [0,1], where (x,y,z) ∈ K = [0,1] × [0,1] × [0,1]; f ∈ C_B (K). By using these operators, define the following positive linear operators on C_B(K): $\begin{matrix} B_{mnk} (f, (x, y, z)) & = (1 + x_{mnk}) B_{mnk} (f, (x, y, z)), \\ (x, y, z) \in K, f \in C (K) . \end{matrix}$ (6.4) Then observe that $B_{mnk} (f_{000}, (x, y, z)) = (1 + x_{mnk}) f_{000} (x, y, z),$ $B_{mnk} (f_{111}, (x, y, z)) = (1 + x_{mnk}) f_{111} (x, y, z),$ $B_{mnk} (f_{222}, (x, y, z)) = (1 + x_{mnk}) f_{222} (x, y, z),$ $B_{mnk} (f_{333}, (x, y, z)) = (1 + x_{mnk}) f_{333} (x, y, z),$ $B_{m n k} (f_{444}, (x, y, z)) = (1 + x_{m n k}) (f_{444} (x, y, z) + \frac{1}{m} (\frac{x}{1 + x} - {(\frac{x}{1 + x})}^{2}) + \frac{1}{n} (\frac{y}{1 + y} - {(\frac{y}{%} 1 + y)}^{2}) + \frac{1}{k} (\frac{z}{1+z} - {(\frac{z}{1+z})}^{2})),$ where $\begin{matrix} B_{mnk} (f_{000}, (x, y, z)) & = 1, B_{mnk} (f_{111}, (x, y, z)) = \frac{x}{1 + x}, \\ B_{mnk} (f_{222}, (x, y, z)) & = \frac{y}{1 + y}, B_{mnk} (f_{333}, (x, y, z)) = \frac{z}{1 + z} \end{matrix}$ and $B_{mnk} (f_{444}, (x, y, z)) = {(\frac{x}{1 + x})}^{2} + {(\frac{y}{1 + y})}^{2} + {(\frac{z}{1 + z})}^{2} .$ Since st₃–lim Ax = 0, we obtain $\begin{matrix} {st}_{3} - lim_{pqj} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{uvw}) - f_{uvw} | |_{C_{B} (K)} = 0 \\ = & {st}_{3} - lim_{pqj} \frac{1}{pqj} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f_{uvw}) - f_{uvw} | |_{C_{B} (K)} = 0 \end{matrix}$ for u,v,w=0,1,2,3. Hence by Theorem 6.3 we conclude that ${st}_{3} - lim_{pqj} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (f) - f | |_{C_{B} (K)} = 0$ for any f ∈ C_B (K).

However, since P-limit and the statistical limit of the triple sequence of Bernstein polynomials of B_mnk (f,x) is not zero, then for u,v,w = 0,1,2,3, ||B_mnk (f_uvw)-f_uvw||_{C_B (K)} is neither P-convergent nor statistically convergent to zero. So, Theorem 6.1 and Theorem 6.2 do not work for our operators defined by (6.4).

Theorem 6.5. Let f be a continuous function defined on the closed interval [0,1]. Let A = (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix and a triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers from [0,1] into itself, $\begin{matrix} {(I_{λ})}_{A}^{g_{\bar{N}}} - lim_{pqj} & \frac{1}{g (λ_{pqj})} \frac{1}{u_{pqj}} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} \cdot \\ B_{mnk} (f, x) - f (x) | | = 0 . \end{matrix}$ (6.5) if and only if $\begin{matrix} {(I_{λ})}_{A}^{g_{\bar{N}}} - lim_{pqj} \frac{1}{g (λ_{pqj})} & \frac{1}{u_{pqj}} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} \cdot \\ B_{mnk} (1, x) - 1 | | = 0 . \end{matrix}$ (6.6) $\begin{matrix} {(I_{λ})}_{A}^{g_{\bar{N}}} - lim_{pqj} & \frac{1}{g (λ_{pqj})} \frac{1}{u_{pqj}} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} \cdot \\ B_{mnk} (v, x) - \frac{x}{1 + x} | | = 0 . \end{matrix}$ (6.7) $\begin{matrix} {(I_{λ})}_{A}^{g_{\bar{N}}} - lim_{pqj} & \frac{1}{g (λ_{pqj})} \frac{1}{u_{pqj}} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} \cdot \\ B_{mnk} (v^{2}, x) - {(\frac{x}{1 + x})}^{2} | | = 0 . \end{matrix}$ (6.8)

Proof. We obtain $E \subset E_{1} ⋃ E_{2} ⋃ \subset E_{3}$ (6.9) and so $\begin{matrix} δ_{{(I_{λ})}_{A}^{g_{\bar{N}}}} (E) \subset δ_{{(I_{λ})}_{A}^{g_{\bar{N}}}} (E_{1}) & + δ_{{(I_{λ})}_{A}^{g_{\bar{N}}}} (E_{2}) \\ + δ_{{(I_{λ})}_{A}^{g_{\bar{N}}}} (E_{3}) . \end{matrix}$ (6.10)

Equations (6.6)-(6.8) give that ${(I_{λ})}_{A}^{g_{\bar{N}}} - \lim_{p q j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot B_{m n k} (f, x) - f (x) ‖ = 0.$

Definition 6.6. Let f be a continuous function defined on the closed interval [0,1] and A = (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular matrix. A triple sequence of Bernstein polynomials (B_mnk(f,x)) is said to be I_λ-weighted statistically convergent of weight g, if for every ∈ > 0 $lim_{pqj} \sum_{m, n, k \in E (u, \in)} a_{i, j, ℓ, m, n, k} = 0,$ (6.11) $E (u, \in) = {\lim_{p q j} \sum_{(m, n, k) = 0}^{p, q, j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} | {(m, n, k) \in I_{p q j} : u_{m n k} | B_{m n k} (f, x) - f (x) | \geq r + \in} | = 0} \in I .$

In this case we write ${(I_{λ})}_{A}^{g_{\bar{N}, u_{p q j}}} - \lim B_{m n k} (f, x) = f (x)$ . Or $B_{m n k} (f, x) \to f (x) {(I_{λ})}_{A}^{g_{\bar{N}, u_{p q j}}}$ .

Theorem 6.7. Let f be a continuous function defined on the closed interval [0,1]. Let A = (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix and a triple sequence of Bernstein polynomials (B_mnk (f,x)) of real numbers from [0,1] which satisfies (6.7)-(6.8) of Theorem 1 and the following condition holds: $lim_{pqj} \frac{1}{g (λ_{pq j})} \frac{1}{u_{pqj}} | | \sum_{(m, n, k) = 1}^{\infty} a_{ij ℓ}^{mnk} B_{mnk} (1, x) - 1 | | = 0 .$ (6.12) Then, $\begin{array}{l} \lim_{p q j} \sum_{(m, n, k) = 0}^{p, q, j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} u_{m n k} \cdot \\ ‖ B_{mnk} (f, x) -f (x) ‖ =0 . \end{array}$ (6.13)

Proof. It follows from (6.12) that $‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} B_{m n k} (1, x) - 1 ‖ \leq C',$ for some constant C' > 0 and for all m,n,k∈ ℕ³. Hence, for f ∈ [0,1], one obtains $u_{m n k} ‖ B_{m n k} (f, x) - f (x) ‖ \leq u_{m n k} (‖ f ‖ ‖ B_{m n k} (f, x) - f (x) ‖ + ‖ f ‖) \leq u_{m n k} C (C' + 1)$ (6.14) Right hand side of (6.14) is constant, so u_mnk|| B_mnk (f,x)-f(x)|| is bounded. Since (6.12) implies (6.6), by Theorem 1 we get that $\begin{array}{l} {(I_{λ})}_{A}^{g_{\bar{N}}} - \lim_{p q j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot \\ B_{mnk} (f, x) -f (x) ‖ =0 . \end{array}$ (6.15)

Definition 6.8. Let f be a continuous function defined on the closed interval [0,1] and A = (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular matrix and let (u_pqj) be a positive nonincreasing sequence. A triple sequence of Bernstein polynomials (B_mnk (f,x)) is said to be I_λ-weighted statistically convergent of weight g, if for every ∈ > 0 $lim_{pqj} \frac{1}{u_{pqj}} \sum_{m, n, k \in E (u, \in)} a_{i, j, ℓ, m, n, k} = 0,$ (6.16) where $E (u, \in) = {\frac{1}{g (λ_{p q j})} {(m, n, k) \in I_{p q j} : u_{m n k} | B_{m n k} (f, x) - f (x) | \geq r + \in}} \in I .$ In this case we write ${(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m n k}) - \lim B_{m n k} (f, x) = f (x)$ . Or $B_{m n k} (f, x) \to f (x) {(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m, n, k})$ as m,n,k → ∈.

Theorem 6.9. Let f be a continuous function defined on the closed interval [0,1]. Let A = (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix. Suppose that (a_mnk), (b_mnk) and (c_mnk) are three positive nonincreasing sequences. Let a triple sequence of Bernstein polynomials (B_mnk (f,x)), (B_mnk (f,y)) and (B_mnk (f,z)) of real numbers from [0,1] such that $B_{mnk} (f, x) - f (x) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m, n, k}),$ $B_{mnk} (f, y) - f (y) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (b_{m, n, k})$ and $B_{m n k} (f, z) - f (z) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (c_{m, n, k}) .$ Then

$(B_{m n k} (f, x) - f_{1} (x)) \pm (B_{m n k} (f, y) - f_{2} (y)) \pm (B_{m n k} (f, z) - f_{3} (y)) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (d_{m, n, k}),$

$(B_{m n k} (f, x) - f_{1} (x)) (B_{m n k} (f, y) - f_{2} (y)) \cdot (B_{m n k} (f, z) - f_{3} (z)) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (d_{m, n, k}),$

$α (B_{m n k} (f, x) - f_{1} (x)) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m, n, k})$ , for any scalar α,where $d_{mnk} = max [a_{mnk}, b_{mnk}, c_{mnk}] .$

Proof. (i) Suppose that $\begin{matrix} B_{mnk} (f, x) - f_{1} (x) & = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m, n, k}), \\ B_{mnk} (f, y) - f_{2} (y) & = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (b_{m, n, k}), \\ B_{mnk} (f, z) - f_{3} (z) & = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (c_{m, n, k}) . . . . . . . . . (*) \end{matrix}$ Given ∈ > 0, define $E^{'} = {\frac{1}{g (λ_{p q j})} {u_{m n k} | (B_{m n k} (f, x) - f_{1} (x)) | \pm (B_{m n k} (f, y) - f_{2} (y)) \pm (B_{m n k} (f, z) - f_{3} (z)) | \geq r + \in}},$ $E^{″} = {\frac{1}{g (λ_{p q j})} {(m, n, k) \in I_{p q j} : u_{m n k} | B_{m n k} (f, x) - f_{1} (x) | \geq r + \frac{\in}{2}}},$ $E^{‴} = {\frac{1}{g (λ_{p q j})} {(m, n, k) \in I_{p q j} : u_{m n k} | B_{m n k} (f, y) - f_{2} (y) | \geq r + \frac{\in}{2}}},$ ${E^{‴}}^{'} = {\frac{1}{g (λ_{p q j})} {(m, n, k) \in I_{p q j} : u_{m n k} | B_{m n k} (f, z) - f_{3} (z) | \geq r + \frac{\in}{2}}} .$ It is easy to see that $E^{'} \subset E^{″} \cup E^{‴} \cup {E^{‴}}^{'} .$ (6.17) This yields that $\begin{matrix} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} \leq \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{''}} a_{i, j, ℓ, m, n, k} \\ + \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'''}} a_{i, j, ℓ, m, n, k} + \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'''';}} a_{i, j, ℓ, m, n, k} \end{matrix}$ (6.18) holds for all (p,q,j) ∈ ℕ. Since d_mnk = max [a_mnk, b_mnk, c_mnk], (6.19) gives that $\begin{matrix} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} \leq \frac{1}{a_{pqj}} \sum_{m, n, k \in E^{''}} a_{i, j, ℓ, m, n, k} \\ + \frac{1}{b_{pqj}} \sum_{m, n, k \in E^{'''}} a_{i, j, ℓ, m, n, k} + \frac{1}{c_{pqj}} \sum_{m, n, k \in E^{''''}} a_{i, j, ℓ, m, n, k} \end{matrix}$ (6.19) Taking limit p,q,j → ∈ in (6.19) together with (*), we obtain $lim_{pqj} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} = 0 .$ (6.20) Thus, $(B_{m n k} (f, x) - f_{1} (x)) \pm (B_{m n k} (f, y) - f_{2} (y)) \pm (B_{m n k} (f, z) - f_{3} (z)) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (c_{m, n, k})$ . Similarly, we can prove (ii) and (iii).

The modulus of continuity of f ∈ [0,1] is defined by $w (f, δ) = \sup {| f (x) - f (y) |, | f (y) - f (z) |, | f (z) - f (x) | : x, y, z \in [0, 1], | x - y |, | y - z |, | z - x | < δ} .$ It is well known that $| f (x) - f (y) |, | f (y) - f (z) |, | f (z) - f (x) | \leq w (f, δ) (\frac{1}{δ} | \frac{x}{1 + x} - \frac{y}{1 + y} | + 1, \frac{1}{δ} | \frac{y}{1 + y} - \frac{z}{1 + z} | + 1, \frac{1}{δ} | \frac{z}{1 + z} - \frac{x}{1 + x} | + 1) .$

Theorem 6.10. Let f be a continuous function defined on the closed interval [0,1]. Let A= (a_{i,j,ℓ,m,n,k}) be a nonnegative RH-regular summability matrix. Let a triple sequence of Bernstein polynomials B_mnk: [0,1] → [0,1] satisfies the conditions

${(I_{λ})}_{A}^{g_{\bar{N}}} - \lim_{p q j} \frac{1}{g (λ_{p q j})} \frac{1}{u_{p q j}} ‖ \sum_{(m, n, k) = 1}^{\infty} a_{i j ℓ}^{m n k} \cdot B_{m n k} (1, x) - 1 ‖ = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (a_{m, n, k}),$

$w (f, λ_{m n k}) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (b_{m, n, k}),$

$w (f, λ_{m n k}) = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (c_{m, n, k})$

with

λ_{m n k} = \sqrt{B_{m n k} (ϕ_{x} : x)}

and

ϕ_{x} (y) = {(\frac{y}{1 + y} - \frac{x}{1 + x})}^{2}, ϕ_{x} (z) = {(\frac{z}{1 + z} - \frac{y}{1 + y} - \frac{x}{1 + x})}^{3}

, where (a_mnk), (b_mnk) and (c_mnk) are three positive nonincreasing sequences, then

| | B_{mnk} (f, x) - f (x) | | = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (d_{m, n, k}),

(6.21)

where d_mnk = max {a_mnk, b_mnk, c_mnk}.

Proof. Consider $\begin{matrix} | B_{mnk} (f, x) - f (x) | \\ \leq B_{mnk} (| f (x) - f (y) |, | f (y) - f (z) |, \\ | f (z) - f (x) |, x) + | f (x) | | B_{mnk} (1, x) - 1 | \end{matrix}$ $\begin{matrix} \leq B_{mnk} (1 + \frac{1}{δ} | \frac{x}{1 + x} |, 1 + \frac{1}{δ} | \frac{y}{1 + y} |, \\ 1 + \frac{1}{δ} | \frac{z}{1 + z} |) w (f, δ) + | f (x) | | B_{mnk} (1, x) - 1 | \\ \leq B_{mnk} (1 + \frac{1}{δ^{2}} | {(\frac{x}{1 + x} - \frac{y}{1 + y})}^{2} |, \\ 1 + \frac{1}{δ^{2}} | (\frac{y}{1 + y} - \frac{z}{1 + z}) |, 1 + \frac{1}{δ^{2}} | {(\frac{z}{1 + z} - \frac{x}{1 + x})}^{2} |) \cdot \\ w (f, δ) + | f (x) | | B_{mnk} (1, x) - 1 | \\ \leq B_{mnk} (1 + \frac{1}{δ^{3}} | {(\frac{x}{1 + x} - \frac{y}{1 + y} - \frac{z}{1 + z})}^{3} |, \\ 1 + \frac{1}{δ^{3}} | (\frac{x}{1 + x} - \frac{y}{1 + y} - \frac{z}{1 + z}) |, \\ 1 + \frac{1}{δ^{3}} | {(\frac{x}{1 + x} - \frac{y}{1 + y} - \frac{z}{1 + z})}^{2} |) w (f, δ) \\ + | f (x) | | B_{mnk} (1, x) - 1 | \\ \leq | B_{mnk} (1, x) - 1 | w (f, δ) \\ + | f (x) | | B_{mnk} (1, x) - 1 | + w (f, δ) \\ + \frac{1}{δ^{2}} B_{mnk} (φ_{x} : x) w (f, δ) + w (f, δ) \\ + \frac{1}{δ^{3}} B_{mnk} (φ_{x} : x) w (f, δ) . \end{matrix}$ Choosing $δ = λ_{m n k} = \sqrt{B_{m n k} (ϕ_{x} : x)}$ , one obtains $\begin{matrix} | | B_{mnk} (f, x) - f (x) | | \leq T | | B_{mnk} (1, x) - 1 | | + 3 w (f, λ_{mnk}) \\ + | | B_{mnk} (1, x) - 1 | | + w (f, λ_{mnk}), \end{matrix}$ (6.22) where T =||f||. For a given ∈ > 0, we will define the following sets: $\begin{array}{l} E' = {(m, n, k) \in ℕ^{3} : \\ u_{m n k} ‖ B_{m n k} (f, x) - f (x) ‖ \geq r + ϵ}, \\ E'' = {(m, n, k) \in ℕ^{3} : \\ u_{m n k} ‖ B_{m n k} (1, x) - 1 ‖ \geq r + \frac{ϵ}{4 T}}, \\ E''' = {(m, n, k) \in ℕ^{3} : \\ u_{m n k} w (f, λ_{m n k}) \geq r + \frac{ϵ}{9}}, \\ E'''' = {(m, n, k) \in ℕ^{3} : \\ u_{m n k} w (f, λ_{m n k}) ‖ B_{m n k} (1, x) - 1 ‖ \geq r + \frac{ϵ}{4}} . \end{array}$ It follows from (6.22) that $\begin{matrix} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} \leq \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{''}} a_{i, j, ℓ, m, n, k} \\ + \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'''}} a_{i, j, ℓ, m, n, k} + \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'''';}} a_{i, j, ℓ, m, n, k} \end{matrix}$ (6.23) holds for all (p,q,j) ∈ ℕ. Since d_mnk = max [a_mnk, b_mnk, c_mnk], (6.23) gives that $\begin{matrix} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} \leq \frac{1}{a_{pqj}} \sum_{m, n, k \in E^{''}} a_{i, j, ℓ, m, n, k} \\ + \frac{1}{b_{pqj}} \sum_{m, n, k \in E^{'''}} a_{i, j, ℓ, m, n, k} + \frac{1}{c_{pqj}} \sum_{m, n, k \in E^{''''}} a_{i, j, ℓ, m, n, k} \end{matrix}$ (6.24) Taking limit p,q,j → ∈ in (6.24), we obtain $lim_{pqj} \frac{1}{d_{pqj}} \sum_{m, n, k \in E^{'}} a_{i, j, ℓ, m, n, k} = 0 .$ (6.25) Hence $‖ B_{m n k} (f, x) - f (x) ‖ = {(I_{λ})}_{A}^{g_{\bar{N}}} - o (d_{m, n, k})$ □

7 Conclusion

In this paper, we have discussed various general topological and algebraic properties of statistical convergent of weight g (A) and also have proved Korovkin approximation theorem by using the notion of weight g (A) limits of a triple sequence space of Bernstein polynomials. All the results will certainly motivate young researchers.

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

Footnotes

Acknowledgement

The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The authors are thankful to the Prof. Dr. Reza Langari, Editor-in-Chief and Prof. Dr. Jianming Zhan, Associate Editor and reviewers of Journal of Intelligent and Fuzzy Systems also Tata realty for finance support.

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