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


 In this paper, we obtain a Korovkin type approximation theorem for power series statistical convergence of functions belonging to the class produced by multivariable modulus of continuity function. As an application of this theorem, we construct a non-tensor product Balázs type BBH operator which does not converge in ordinary sense. Moreover, we study promised approximation properties of this operator and compute the rate of convergence. Finally, we prove that our new approximation result works but its classical case fails.

ABSTRACT. In this paper, we obtain a Korovkin type approximation theorem for power series statistical convergence of functions belonging to the class produced by multivariable modulus of continuity function. As an application of this theorem, we construct a non-tensor product Balázs type BBH operator which does not converge in ordinary sense. Moreover, we study promised approximation properties of this operator and compute the rate of convergence. Finally, we prove that our new approximation result works but its classical case fails.
c 2022 Mathematical Institute Slovak Academy of Sciences

Preliminaries
Summability theory has many applications in probability limit theorems, approximation theory and differential equations etc. ( [8,17,19,30]). In Korovkin type approximation theory [2], by using summability methods may be beneficial when a sequence of positive linear operators does not converge to the identity operator in ordinary sense. In this direction, the first Korovkin type theorem was given by considering the concept of statistical convergence by Gadjiev and Orhan in [17]. Following that study many authors gave several approximation results via summability theory (see, e.g., [3,11,12,21,27,28,31]). Let x = (x j ) be a real sequence and let A = (a nj ) be a summability matrix. If the sequence {(Ax) n } is convergent to a real number L, then we say that the sequence x is A-summable to the real number L where the series (Ax) n := ∞ j=0 a nj x j is convergent for any n ∈ N 0 and N 0 = {0, 1, . . . }. A summability matrix A is said to be regular if lim(Ax) n = L whenever lim x = L (see, [10]).
Let A = (a nj ) be a non-negative regular summability matrix and let E ⊂ N 0 . Then the number is said to be the A-density of E whenever the limit exists (see, [20] matrix, then δ(E) := δ C (E) is called the (natural or asymptotic) density of E (see, [15]) where C = (c nj ) is the summability matrix defined by A real sequence x = (x j ) is said to be A-statistically convergent (see, [16]) to a real number L if for any ε > 0, δ A ({j ∈ N 0 : |x j − L| ≥ ε}) = 0. In this case we write st A -lim x = L. If we consider the Cesàro matrix, then C-statistical convergence is called statistical convergence [14]. In general, A-statistical convergence is regular and there exists some sequences which are A-statistically convergent but not ordinary convergent.
In what follows we recall the concept of power series method [10]: Let (p j ) be a real sequence with p 0 > 0 and p 1 , p 2 , . . . ≥ 0, and such that the corresponding power then we say that x = (x j ) is convergent in the sense of power series method P p . Such a summability method is a general version of the Abel and Borel summability methods. Related to these methods, some approximation results can be found in [5,9,23,24,26,27,29]. The following theorem characterizes the regularity of a power series method. Unver 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 N 0 , we mention it as below: 31]). Let P p be a regular power series method and let E ⊂ N 0 . 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.
. 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 ∈ N 0 : |x j − L| ≥ ε}) = 0 for any ε > 0. In this case we write st Pp -lim x = L.
In 1999, Gadjiev and Ç akar [18] gave a Korovkin type theorem in ordinary sense, by using the test functions t 1+t v 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 Ç akar considering A-statistical convergence in multivariate case and showed that the new theorem is quite useful. Aktuglu andÖzarslan [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 H m ω 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 Furthermore, we simply write f (x) rather f (x 1 , . . . , x m ) for x = (x 1 , . . . , x m ) ∈ S, we also write a n x rather (a 1,n x 1, a 2,n x 2 , . . . , a m,n x m ). 1 denotes a function such that f (x 1 , . . . , x m ) = 1 and, for any x = (x 1 , . . . , x m ) , y = (y 1 , . . . , 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.
Let H m ω (S) denote the space of all real valued functions defined on S satisfying for all x = (x 1 , . . . , x m ) , y = (y 1 , . . . , y m ) ∈ S. It is easy to see that H m ω (S) ⊂ C B (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 H m ω (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 H m ω (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.
Throughout the paper, we assume that L 0 (f ) = 0 for any f ∈ C B (S) and we use the following test functions:ê 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 ) p n := 0, n = 2k, 1, n = 2k + 1, and we consider the sequence (b n ) such as b n := n, n = 2k, 1, n = 2k + 1, it is not convergent, but P p -statistical convergent.

Main result
In this section, using P p -statistical convergence, we prove a Korovkin type theorem for the operators (1.2) . Now, we recall, the following Korovkin type theorem which was given by Gadjiev and Ç akar [18]. 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.

BALÁZS TYPE BLEIMANN, BUTZER AND HAHN OPERATORS
From the boundedness of f on S, one has . . , 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 . For a given s > 0 we select > 0 such that < s. Let us define the following sets , i = 0, 1, 2, . . . , m + 1.
Then, by (2.1) , we have U ⊂ m+1 i=0 U i . Hence, for all n ∈ N, which completes the proof.

Power series statistical convergence of the operators (L n )
In this section, we investigate the power series statistical convergence properties of the operators (1.2) and compute the rate of the P p -statistical convergence by means of modulus of continuity [4]. Before studying the promised approximation properties of these operators, we give the following lemma which can be proved as in [25].
In the following theorem, we investigate the power series statistical convergence properties of the sequence (L n ) for any f ∈ H m ω (S) on S. 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 > 0, It follows that Taking into account the inequalities we get where λ is the greatest integer of λ.

Concluding remarks
It should be remarked that Theorem 2.1 and multivariate extension of this theorem given in [25] do not work for the operators (1.2), but Theorem 2.2 works. So we can demonstrate that Theorem 2.2 is more powerful than Theorem 2.1.