A STANCU TYPE EXTENSION OF CHENEY AND SHARMA OPERATORS

. In this paper, we introduce a Stancu type extension of the well known Cheney and Sharma operators. We consider a recurrence relation for the moments of the operators and give a local approximation result via suitable K -functional. Moreover, we show that each operator preserves the Lipschitz constant and order of a given Lipschitz continuous function.

In [8], Stancu constructed the following Bernstein type linear positive operators r is a non-negative integer parameter with n > 2r (see, also, [9]).
In [15], Yang, Xiong and Cao extended the operators L n,r given by (10) to the multivariate setting on a simplex and called them as multivariate Stancu operators. In the work, using elementary method, the authors proved that the multivariate Stancu operators preserves Lipschitz property of the operand. In [4], Bustamante and Quesada gave an asymptotic property for Stancu operators L n,r related to Voronovskaja-type formula.
In the present paper, we consider Stancu operators L n,r in the basis of the Bernstein type Cheney and Sharma operators G β n given by (5). For this purpose, we consider for f ∈ C[0, 1] and r is a non-negative integer parameter with n > 2r, n ∈ N, where P β n−r,k is given by (6) with n − r in places of n. We shall call these operators as Stancu type extension of Cheney and Sharma operators. For the calculation of moments, we use the same recurrence relationship which is obtained from another quantity that is slightly different from (9). Namely, the quantity (9) is closely related to (1), whereas the quantity that we shall use is related with (3). We study local approximation with the help of suitable K−functional, and show the preservation of Lipschitz' constant and order of a Lipschitz continuous function by L β n,r . To get approximation results, as in [5], we take β, as a sequence of positive real numbers such that β = o( 1 n ) (n → ∞). It is obvious that L 0 n,r reduces to the Stancu operator L n,r given by (10) and L β n,0 reduces to the Cheney and Sharma operator (5).

AUXILIARY RESULTS
Using the similar technique of [5], we consider the following quantity T (k, n, x, y) to get the subsequent recurrence relationship.
Then one has Namely, T satisfies the same reduction formula that (9) holds.
Proof. Direct calculation gives the result.
Conclusion 2.1. For the quantity (12), one has the following results: (iii) Recursive application of the formula (13) gives that Moreover, as in [5], using the fact v! = ∞ 0 e −s s v ds and the binomial formula, one gets (14) yT Below, making use of (12) and (13), we give G β n (e 2 ; x).
Proof. It is easy to see that Using (14), G β n (e 2 ; x) can be represented as where A n and B n are given by (15) and (16) we have As in [9], using the inequalities In view of (17), we obtain Hence, using the fact β = β n > 0 (n ∈ N) satisfying lim n→∞ nβ n = 0, we conclude that lim n→∞ A n = 1 and lim n→∞ B n = 1.

APPROXIMATION PROPERTIES OF THE STANCU TYPE EXTENSION OF CHENEY AND SHARMA OPERATORS
In this section, we study some approximation properties of the Stancu type extension of Cheney and Sharma operators L β n,r given by (11). The moments of the operators can be expressed in terms of the moments of the Cheney and Sharma operators G β n . Namely, we have x (x + 2β) A n−r + x (n − r − 2) β 2 B n−r + x n 2 (n − r) (1 + 2xr) + r 2 , where A n−r and B n−r are given by (15) and (16), respectively.
Thus, from Lemma 3.1 one easily obtains the following result.
Let us denote the uniform norm on C [0, 1] by . . It is not difficult to show that the operators L β n,r are bounded from C [0, 1] onto itself: Recall that the Peetre K-functional is defined as where δ > 0, W 2 := {g ∈ C [0, 1] : g , g ∈ C [0, 1]}. From p.177, Theorem 2.4 of [7], there is a positive constant C > 0 such that where is the second order modulus of smoothness of f ∈ C [0, 1] .
, n > 2r, r ∈ N∪{0} , n ∈ N, and β be a sequence of positive real numbers such that β = o 1 n . Then , where δ n,r (x) is the same as in Corollary 3.1 and C is a positive constant.
Proof. For any function g ∈ W 2 and x, t ∈ [0, 1], the Taylor formula gives that Applying L β n,r on both sides of the above formula, linearity and Lemma 3.1 give that Therefore, Passing to the infimum over all g ∈ W 2 and taking (18) into consideration, we obtain which completes the proof.
Next result provides the property of the preservation of Lipschitz' constant and order of a Lipschitz continuous function by each L β n,r . The same result for the Bernstein polynomials was proved by Brown, Elliott and Paget [3], also, for the Cheney and Sharma operators G β n was obtained in [2] and for the multivariate Stancu operators was proved in [15].
Recall that the class Lip M (α, [0, 1]) and the convexity of f ∈ C[0, 1] are defined, respectively, as where α 1 , α 2 , . . . , α n ≥ 0 satisfying α 1 + α 2 + · · · + α n = 1. Proof. Assume that x, y ∈ [0, 1] satisfy y ≥ x. Following similar steps used in [2], from (11) and (3) L β n,r (f ; y) can be written as Changing the order of the summations and letting j − k = l in the result, we obtain Thus, subtracting (20) from (19) we get Adding and dropping the terms yf ( k n ) and xf ( k+l+r n ), the above formula takes to the following form: Therefore, taking the absolute values of both sides of the last formula, and using the hypothesis that f ∈ Lip M (α, [0, 1]) in the result, we get by the assumption that x ≤ y. Changing the order of the summations, the above formula gives that Here, taking x 1 = l n , x 2 = l+r n , and regarding the nonnegative constants α 1 , α 2 as α 1 = y − x and α 2 = 1 − (y − x) that satisfy α 1 + α 2 = 1, and using the fact that g (t) = t α , 0 < α ≤ 1, is concave, then the last formula reduces to where P β n−r is given by (6). Here, the case α = 1 is obvious. For the case 0 < α < 1; application of Hölder's inequality, with conjugate pairs p = 1 α and q = 1 1−α , leads to