A -STATISTICAL CONVERGENCE FOR A CLASS OF POSITIVE LINEAR OPERATORS

. In this paper, we introduce a sequence of positive linear operators deﬁned on the space C [0 , a ] (0 < a < 1) , and provide an approximation theorem for these operators via the concept of A -statistical convergence. We also compute the rates of convergence of these approximation operators by means of the ﬁrst and second order modulus of continuity and the elements of the Lipschitz class. Furthermore, by deﬁning the generalization of r -th order of these operators we show that the similar approximation properties are preserved on C [0 , a ] .


INTRODUCTION
The concept of a limit of a sequence has been extended to statistical limit ( [15], [17], [18]) by using the natural density δ of a set K of positive integers: δ(K) := lim n 1 n {the number k ≤ n such that k ∈ K} whenever the limit exists (for the natural density, see [27]).A sequence x = (x k ) is said to be statistically convergent to a number L if for every ε > 0, δ{k : |x k − L| ≥ ε} = 0 and it is denoted by st − lim k x k = L. Let A = (a jn ), j, n = 1, 2, ..., be an infinite summability matrix.The A-transform of the sequence x, denoted by Ax := {(Ax) j }, is given by (Ax) j := ∞ n=1 a jn x n provided the series converges for each j.A is said to be regular if lim j (Ax) j = L whenever lim j x j = L [4].Assume that A is a nonnegative regular summability matrix.The A-density of K, denoted by δ A (K), is defined by δ A (K) := lim j ∞ n=1 a jn χ K (n) provided the limit exists, where χ K is the characteristic function of K.Then, x = (x n ) is said to be A-statistically convergent to a number L if, for every ε > 0, δ A {n ∈ N : |x n − L| ≥ ε} = 0; or equivalently lim j n: |xn−L|≥ε a jn = 0. We denote this limit by st A − lim x = L (see [16], [19], [23], [26]).The case in which A = C 1 , the Cesáro matrix of order one, reduces to the statistical convergence, and also if A = I, the identity matrix, then it coincides with the ordinary convergence.We note that if A = (a jn ) is a nonnegative regular summability matrix such that lim j max n {a jn } = 0, then A-statistical convergence is stronger than convergence [23].
Chlodovsky [9] was the first to notice that the Bernstein polynomials, converge to middle of jump at the point of simple discontinuity of a function.That is, if x is a point of discontinuity of first kind, then But this phenomenon does not always take place for general positive linear approximation operators.An example was given by Bojanic and Cheng in [5] where they showed that the Hermit-Fejer interpolation operator, where the nodes x n,k = cos( (2k−1)π 2n ) are the zeros of Chebyshev polynomials T n (x) = cos(n cos −1 x), does not converge at a point of simple discontinuity.However, Bojanic and Khan [6] showed that the Cesáro averages of the Hermit-Fejer operator, 1 n n k=1 H k (f ; x), do converge to the mid point of the jump discontinuity.So, it shows that this summability method is stronger than the classical sense in approximation theory.In recent years another form of regular summability transformation has shown to be quite effective in "summing" nonconvergent sequences which may have unbounded subsequences (see [16], [17]).Furthermore, some Korovkin type approximation theorems have been studied via statistical convergence and A-statistical convergence in [11], [12], [13], [20].
The aim of the present paper is to provide an A-statistical approximation theorem for Agratini type operators [1].Note that Agratini's operators are a Stancu type generalization (see [29]) of the operators in [10].We also give the rates of A-statistical convergence of these operators by means of the first and second order modulus of continuity and the elements of the Lipschitz class.

STATISTICAL APPROXIMATION OF POSITIVE OPERATORS
In this section, we give a generalization of Agratini's operators [1] and obtain an approximation theorem for these operators by using A-statistical convergence.
We now introduce the sequence of operators Ω n on C[0, a], the space of all continuous functions on [0, a], by Now we analyze our operators Ω n and give their applications in approximation theory settings.To obtain that we first assume given by (2.5) turn out to be unmodified Meyer-König and Zeller Operators [25] M Choosing Taking x−1 for t ≤ 0, then the operators Ω n coincide with the operators of Khan [21] where L n k (t) denotes the Laguerre polynomial of degree k defined by We should note that choosing ρ n,k = k + n in (2.4) and replacing the matrix A by the identity matrix I conditions (2.1)-(2.4)reduce to all those in [1].
It is easy to see that each Ω n is positive and linear, and also Ω n (1; x) = 1 (for every n ∈ N) holds.
Throughout the paper we denote the usual norm of the space C[0, a] by To construct our A-statistical approximation theorem for the sequence {Ω n } we need the following two lemmas whose proofs can immediately be obtained with the similar methods used in [11] and [14].Lemma 2.2.Let A = (a jn ) be a nonnegative regular summability matrix.Then we have Combining Lemmas 2.1 and 2.2 we have the following main result.
Theorem 2.3.Let A = (a jn ) be a nonnegative regular summability matrix.Then, for all f ∈ C[0, a], we have Proof.By Lemmas 2.1 and 2.2, we immediately get So, the result follows from Theorem 1 in [20], (see also [12]).We note that Theorem 1 in [20] is given for statistical convergence, but the proof also works for A-statistical convergence.
When the matrix A is replaced by the identity matrix I in Theorem 2.3, then the following result holds at once.Corollary 2.4.For all f ∈ C[0, a], the sequence {Ω n (f )} converges uniformly to f on [0, a].
The following example shows that A-statistical approximation in Theorem 2.3 is stronger than ordinary norm-wise convergence in Corollary 2.4.
Let A = C 1 = (c jn ), the Cesáro matrix of order one, defined by Then C 1 -statistical convergence is known as statistical convergence [15].Assume that (u n ) is defined by Observe that (u n ) is non-convergent, but it is statistically convergent to 1, i.e., st − lim n u n = 1.Let {Ω n } be the sequence of positive linear operators given (2.5).Now define the operators and hence Since, for all f ∈ C[0, a], the sequence {Ω n (f )} converges uniformly to f and also (u n ) converges statistically to 1, it follows from (2.6) that Hence Theorem 2.3 holds for the operators Ω * n .However, since (u n ) is nonconvergent, the sequence {Ω * n (f )} is not uniformly convergent to f, which does not satisfy Corollary 2.4.

RATES OF A-STATISTICAL CONVERGENCE
In this section, we compute the rates of A-statistical convergence in Theorem 2.3 by means of the first and second order modulus of continuity and the elements of the Lipschitz class.
It is well known that for any δ > 0 and each t, x ∈ [0, a] The next result gives the rate of A-statistical convergence of the sequence {Ω n (f )} (for all f ∈ C[0, a]) in Theorem 2.3 by means of first order modulus of continuity.
Theorem 3.1.For all f ∈ C[0, a], we have Proof.We will use Popoviciu's technique given in [28].Let f ∈ C[0, a].By linearity and monotonicity of Ω n we obtain By (3.1) and the Cauchy-Bunyakowsky-Schwarz inequality we have This implies that A n (x) For each x ∈ [0, a], one can write So, by (2.9) and (2.12) we get (3.5)sup where whence the result.
Let f ∈ C[0, a].Then the second order modulus of continuity of f denoted by w 2 (f, δ) is defined as This modulus is also known as Zygmund's modulus for the function f.
In order to estimate this order of approximation via second modulus of continuity we will benefit the Peetre's K-functional.Now we denote the space of the functions f such that f, f , f ∈ C[0, a] by C 2 [0, a] and define the following norm in the space C 2 [0, a] by Then the following Peetre's K-functional [3] (see also [7]) is given by where B 3 and δ n are the same as in (3.2).
Proof.If g ∈ C 2 [0, a] then we have Applying the operator Ω n to the (3.8) we get . On the other hand, since Ω n is a linear operator, we have Thus, by using Ω n (1; x) ≡ 1 and (3.9), we can write After some simple calculations, using (3.4) in (3.10) we can write By taking infimum over g ∈ C 2 [0, a] on the both sides of ( The following theorem estimates the rate of convergence of the sequence {Ω n } to the function f via Zygmund modulus. where B 3 and δ n are the same as in (3.2).
We will now study the rate of A-statistical convergence of the positive linear operators Ω n by means of the elements of the Lipschitz class Lip M (α), where M > 0 and 0 < α ≤ 1.
We recall that a function f ∈ C[0, a] belongs to Lip M (α) if the inequality holds.
Theorem 3.4.For all f ∈ Lip M (α), we have where B 3 and δ n are the same as in (3.2).
We consider the following generalization of the positive linear operators Ω n defined by (2.5) where f ∈ C [r] [0, a], r = 0, 1, 2, ..., and n ∈ N. We call the operators (4.1) the r-th order of the operators Ω n (see, for instance, [10], [22]).Note that taking r = 0 we get the sequence {Ω n } defined by (2.5).Now we have the following where B(α, r) is the beta function and r, n ∈ N.
Proof.By (4.1) we get It is known from Taylor's formula that 3 )w(g, δ n ), where B 3 and δ n are the same as in (3.2) and g is defined by (4.8).

Lemma 2 . 1 .
Let A = (a jn ) be a nonnegative regular summability matrix.Then we have st A − lim n Ω n (t; x) − x = 0, where .C[0,a] denotes the ordinary sup norm on the space C[0, a].