A Korovkin Type Approximation Theorem For Balázs Type Bleimann, Butzer and Hahn Operators via Power Series Statistical Convergence

Definition 1 ([10]).

A power series method P_p is regular if and only if for any j ∈ ℕ₀

Ünver and Orhan [31] introduced the concept of P_p-statistical convergence. This type of convergence is based on the notion of the P_p-density of a subset of ℕ₀, we mention it as below:

Definition 2 ([31]).

Let P_p be a regular power series method and let E ⊂ ℕ₀. If

exists, then δ_Pp (E) is called the P_p-density of E. Note that δ_Pp (E) satisfies the inequality 0 ≤ δ_Pp(E) ≤ 1, whenever it exists.

Definition 3 ([31]).

Let x = (x_j) be a real sequence and let P_p be a regular power series method. Then x is said to be P_p-statistically convergent to L if for any ε > 0

i.e., δ_Pp ({j ∈ ℕ₀ : |x_j − L| ≥ ε}) = 0 for any ε > 0. In this case we write st_Pp-lim x = L.

In 1999, Gadjiev and Çakar [18] gave a Korovkin type theorem in ordinary sense, by using the test functions t1+tv for v = 0, 1, 2. In that research, the authors also investigated uniform approximation properties of the Bleimann, Butzer and Hahn (BBH) operators. Later, Erkuş and Duman [13] studied an extention of the Korovkin type theorem given by Gadjiev and (Çakar considering A-statistical convergence in multivariate case and showed that the new theorem is quite useful. Aktuğlu and Özarslan [1] proved a Korovkin type theorem by using ideal convergence. They also introduced multivariate BBH type operators and proved ideal convergence of sequence of these operators which doesn’t converge in ordinary sense.

In the present paper, taking into account the P_p-statistical convergence, we obtain a Korovkin type approximation theorem in multivariable case. We also construct non-tensor multivariete Balázs type BBH operator {L_n} which does not converge in ordinary sense. Furthermore, we prove the P_p-statistical convergence of these operators for the function f belonging the space Hωm which is a subspace of continuous and bounded functions defined on S. Finally, we compute the rate of the P_p-statistical convergence of the operators {L_n} by means of modulus of continuity function.

Now, we recall some notations of multivariate setting:

Let us consider the set S ⊂ ℝ^m, (m ∈ ℕ), given by

For x = (x₁, …, x_m) ∈ S and k = (k₁, …, k_m) ∈ ℕ^m ∪ {0}, and n ∈ ℕ, we denote

The Euclidean norm of x = (x_1,, …, x_m) is given by ||x||=∑i=1mxi2.

Furthermore, we simply write f (x) rather f (x₁, …, x_m) for x = (x₁, …, x_m) ∈ S, we also write a_nx rather (a_1,nx_1,a_2,nx₂, …, a_m,nx_m). 1 denotes a function such that f (x₁, …, x_m) = 1 and, for any x = (x₁, …, x_m), y = (y₁, …, y_m) ∈ S, x ≤ y means that x_i ≤ y_i for each i = 1, 2,…, m.

Let C_B (S) denote the space of all real valued continuous and bounded functions defined on S, with norm

Below, we give the definition of the modulus of continuity function.

Definition 4 ([4]).

A non-negative function ω (u) defined in S ⊂ ℝ^m is called a function of modulus of continuity, if it satisfies the following conditions for any δ = (δ₁, …, δ_m), μ = (μ₁, …, μ_m) ∈S :

• (1) ω (δ) is continuous for all δ_i, i = 1, …, m,

• (2) ω (0) = 0, where 0 = (0, 0, …, 0),

• (3) ω (δ) = ω (δ₁, …, δ_m) is non-decreasing, i.e., ω (δ) ≥ ω (μ) for δ ≥ μ,

• (4) ω (δ) is sub-additive, i.e., ω (δ + μ) ≤ ω (δ) + ω (μ).

Let Hωm(S) denote the space of all real valued functions defined on S satisfying

for all x = (x₁, …, x_m), y = (y₁, …, y_m) ∈ S. It is easy to see that Hωm(S)⊂CB(S). Letting univariate modulus of continuity in (1.1), we obtain the space H_ω defined in [18].

In [25], Söylemez et al. used the class Hωm(S) to prove a Korovkin type theorem for uniform convergence. As an application of this theorem, the authors gave uniform approximation properties of non-tensor multivariate BBH operators. They also show that uniform convergence does not achieve when selecting a function from the space C_B (S), instead of Hωm(S).

In [22], Özarslan et al. gave a Korovkin type theorem in multivariable case for Balázs-type BBH operators considering a class which was produced by univariate modulus of continuity function. In that research, the authors also obtained rate of the convergence for Bivariate case.

For f ∈ C_B (S), we consider the following Balázs-type BBH operator which is not a tensor product setting.

where x = (x₁, x₂, …, x_m) ∈ S, b_n ≥ 0, a_i,n ≥ 0 for all i = 1, 2, …, m, and n ∈ ℕ.

Throughout the paper, we assume that L₀ (f) = 0 for any f ∈ C_B (S) and we use the following test functions:

The following example shows that there exists a sequence (b_n) such that P_p-statistical convergence holds but ordinary convergence does not hold.

Example 1.

We assume that P_p is the regular power series method with (p_n)

and we consider the sequence (b_n) such as

it is not convergent, but P_p-statistical convergent.

If we take m = 1, b_n = 1, n ∈ ℕ, the operators (1.2) reduce to the Balázs type Bleimann Butzer and Hahn operators (see; [6, 7]).

Theorem 2.1 ([18]).

Let (A_n) be a sequence of positive linear operators fromH_ω → C_B [0, ∞). Then we have

Multivariate extension of Theorem 2.1 was proved in [25]. We remark that the sequence of the operators (1.2) does not satisfy the conditions of these two theorems. Therefore, it may be beneficial to use the following theorem.

Theorem 2.2.

LetP_pregular power series method and let {R_n (f)}_n∈ℕbe a sequence of linear positive operators fromHωm(S) to C_B (S). If

Proof. It is clear that (2.2) implies (2.1). Suppose that f∈Hωm(S) and x = (x₁, …, x_m), t = (t₁, …, t_m) are any two elements of S. Then, from Definition 4, for any given ϵ > 0, we can find a δ_i > 0, for i = 1, 2, …, m and taking δ = min {δ₁, δ₂, …, δ_m}, we may write

On the other hand, if ti01+|t|−xi01+|x|≥δ for some i₀ ∈ {1, 2, …., m}, then we have

From the boundedness of f on S, one has

whenever ti1+|t|−xi1+|x|≥δ for some i₀ ∈ {1, 2, …, m}. Therefore, for all x, t ∈S we can write

Application of the operators (R_n) to (2.4) gives

From the linearity and positivity of the operators (R_n) and considering (2.1), we have

which implies

where K=maxϵ+‖f‖CB+2m‖f‖CBδ2,4‖f‖CBδ2.

For a given s > 0 we select ϵ > 0 such that ϵ < s. Let us define the following sets

Then, by (2.1), we have U⊂∪i=0m+1Ui. Hence, for all n ∈ ℕ,

which completes the proof. □

Lemma 3.1.

In the following theorem, we investigate the power series statistical convergence properties of the sequence (L_n) for any f∈Hωm(S) on S.

Theorem 3.1.

Let (L_n) be the sequence of the operators defined by (1.2) and suppose thatstPp-limnbn=1, stPp-limnai,n=1, for alli = 1, 2, …, m. Then for anyf∈Hωm(S)we have

Proof. From Theorem 2.2, it suffices to show that (2.1) holds for (L_n). Indeed, from Lemma 3.1 and the hypothesis, we have

Also, for all i = 1, 2, …, m, we can write

Thus, we reach to

for all i = 1, 2, …, m. Now, let us define the following sets for any ϵ > 0,

for i = 1, …, m, it is obvious that Ni⊂Ni1∪Ni2. Therefore, we can write

for i = 1, …, m. By the assumptions and stPp-limnnn+1=1, we have

On the other hand, from (3.3), we can write

It follows that

Taking into account the inequalities

we get

Also, we have

Therefore, we reach to

Now, we define the following sets for any ϵ > 0 that

we easily see that K⊂∪i=1mKi1∪Ki2∪Ki3∪Ki4. Finally we have

Since stPp-limnbn=1, stPp−limnai,n=1 for all i = 1, 2, …, m and stPp-limnn(n−1)(n+1)2=1, we get

Thus, the proof is completed. □

Now, we calculate the rate of P_p-statistical convergence via modulus of continuity function. The modulus of continuity function ω given in Definition 4 satisfies the inequality

where ⌊λ⌋ is the greatest integer of λ.

Here, we are ready to give the following theorem:

Theorem 3.2.

Let (L_n) be a sequence of positive linear operators defined by (1.2). If

• (i) stPp-limnLne^0−1CB=0,

• (ii) stPp-limnωδn=0,

Proof. Selecting λ=t1+|t|−x1+|x|δ, δ > 0 in (3.4), for any f∈Hωm(S), we obtain

where ω (δ) such that δ_i = δ, for i = 1, 2, …, m ( that is; ω (δ) = ω (δ, δ, …, δ)). From (3.1), we obtain

Now, if we take δn=supx∈SLnt1+|t|−x1+|x|2;x12, then we get

From (i) and (ii), we have

which completes the proof. □