Approximation properties of the modified Lupas-Kantorovich type operators

In this paper, the author introduce a class of modified Lupas-Kantorovich type operators which preserve constant and linear functions. By using modulus of continuity, modulus of smooth, K-functional and lipschitz class, the rate of convergence of these operators are derived. Finally, the author present a voronovskaya-type asymptotic formula.


Introduction
In the year 1987, Lupas [1] introduced Polya-Bernstein operators where In [2], Miclaus studied some approximation properties of Bernstein-Stancu type operators based on Polya distribution.Recently, Gupta and Rassias [3] introduced the Durrmeyer variant of the operators (1) as follows: where p n,k (t) = ( n k )t k (1 − t) n−k .In [4], to approximate Lebesgue integrable functions, Agrawal introduced the following integral modification of the operators (1): In 2003, King [5] introduced the modified Bernstein operators.Later, this idea was applied to some other well-known approximating operators,such as, the Szasz-Mirakjan operators [6], the Baskakov operators [7], the q-operators [8].Obviously, these operators have a better rate of convergence than the classical operators.
Inspired by the idea of King, we propose the modified variant of the operators (3) in the following way: , 3  4 ].The purpose of this paper is to study local approximation, pointwise estimates and global approximation results.Finally, we present a voronovskaya-type asymptotic formula.
We get some definition as follows: e-mail: lianboyong@163.comDefinition 2 (the second order modulus of smoothness) where , the first order modulus of smoothness is given by [9] , is the class of all absolutely continuous functions on [0, 1].

Some Lemmas
We start this section with the following uesful lemmas, which will be used in the sequel.
Using the same method to lemma 2, we obtain Lemma 3 For e i = t i , i = 0, 1, 2, x ∈ [ 1 4 , 3  4 ], we have 2 .Remark 1 By simple applications of Lemma 3, we get Remark 2 When n sufficient large, we have Proof From the definiton (4) and Lemma 3, we have Lemma 6(see [9])

Conclusion
Proof From the Lemma 3, we get By Korovkin theorem, it follows that Proof By Taylor's expansion, we may write where φ(t; x) is the Peano form of the remainder, φ(t; x) ∈ C[0, 1] and lim t→x φ(t; x) = 0.
Theorem 3 For f ∈ C[0, 1], x ∈ [ 1 4 , 3  4 ], when n sufficient large, we have (15) Proof Using linearity and monotonicity of D (1/n) n ( f, x) we easily get, for every λ > 0, that , when n sufficient large, there exists a constant C > 0 such that . By Taylor's expansion, we get From Lemma 3, we have Taking the infimum on the right hand side over all g ∈ W 2 , we obtain By Lemma 5, we get 3  4 ] and φ(x) = √ x(1 − x), when n sufficient large, there exists a constant C > 0 such that Proof Using the representation we get For any x, t ∈ (0, 1), we find that Using Cauchy-Schwarz inequality, we obtain Taking the infimum on the right hand side over all g ∈ W φ [0, 1], we obtain By Lemma 6, we get Theorem 6 For f ∈ Lip M (β), x ∈ [ 1 4 , 3  4 ], when n sufficient large, we have Proof By Hölder's inequality with p =