Approximation by modified Kantorovich–Stancu operators

In the present paper, we study a new kind of Kantorovich–Stancu type operators. For this modified form, we discuss a uniform convergence estimate. Some Voronovskaja-type theorems are given.


Auxiliary results
(ii) We have that Corollary 2.2 For any p ∈ N * , there exists a constant C(p), independent of m and x, such that for every x ∈ [0, 1].
Proof First we have The following inequality where c(i) is a constant independent of m, can be found in [16] for mX ≥ 1, X = x(1x) and in [5] for mX < 1. Taking From (2.4) and Lemma 2.1, we obtain estimate (2.1).
The first four central moments for K (α,β) m are as follows: , (ii) For any p ≥ 1 and x ∈ (0, 1), there exists a positive constant B(p) independent of m and x such that where · is the uniform norm on [0, 1]. (iii) From (i) and (ii) it follows Remark 2.4 From Mamedov's theorem [10] it follows that: If p ∈ N * is even and f ∈ C p ([0, 1]), for any x ∈ [0, 1], we have that
In order to prove the uniform convergence of the operators K (α,β) m , we give the Korovkin theorem: For the first case, we obtain the following result: In the second case, we have the following: (3.5) Using the remarks for case 2, it follows that the operators K where l i = lim m→∞ a i (m), i = 0, 1.
uniformly on [0, 1]. Combining these two results, the proof is finished.
In what follows, we will denote by ω(f ; ·) the first order modulus of continuity of the function f Theorem 3.8 Let a 0 (m), a 1 (m) be two bounded sequences that verify (3.2).
Proof By (3.1), we have that We need an upper bound for a(x; m) and a (1x; m). Note that this is the same upper bound for both. From   in (3.11), we obtain (3.8).