Higher order Kantorovich-type Szász–Mirakjan operators

In this paper, we define new higher order Kantorovich-type Szász–Mirakjan operators, we give some approximation properties of these operators in terms of various moduli of continuity. We prove a local approximation theorem, a Korovkin-type theorem, and a Voronovskaja-type theorem. We also prove weighted approximation theorems for these new operators.


Introduction and auxiliary results
The well-known Bernstein polynomials belonging to a function f (x) defined on the interval [0, 1] are defined as follows: where p n,k (x) = e -nx (nx) k k! .
An operator L : C[0, 1] → C[0, 1] is said to be convex of order l -1 if it preserves convexity of order l -1, l ∈ N, where N is the set of natural numbers. The classical Bernstein operator is an example of a mapping convex of all orders l -1, l ∈ N. For an operator L being convex of order l -1, consider 1] given by Q l may be considered as an lth order Kantorovich modification of L. The construction of positive operators Q l , l ≥ 0, is most useful in simultaneous approximation where for appropriate mappings L the difference is considered (see [6,7,11]). On the other hand, we know that Kantorovich-type Szász-Mirakjan operators can be defined as follows: By using the lth order integral and the above definition of the Kantorovich-type Szász-Mirakjan operators, we define a new lth order Kantorovich-type Szász-Mirakjan operator as follows: The paper is organized as follows. In the preliminaries section we give some known results and we derive a recurrence formula for the lth order Szász-Mirakjan-Kantorovich operators K l n (f ; x). With the help of the derived recurrence formula, we calculate the moments K l n (t m ; x) for m = 0, 1, 2, 3, 4 and we calculate the central moments K l n ((tx) m ; x) for some m. In Sect. 3, we prove a local approximation theorem, a Korovkin-type approximation theorem, and a Voronovskaja-type theorem. We obtain the rate of convergence of these types of operators for Lipschitz-type maximal functions, second order modulus of smoothness and Peetre's K -functional. In Sect. 4, we investigate weighted approximation properties of the lth order Szász-Mirakjan-Kantorovich operators in terms of the modulus of continuity.

Preliminaries
We consider the following class of functions. Let In the following lemma we give the moments of the Szász operator up to the fourth order.

Lemma 1 ([23]) We have
In the following lemma we derive a recurrence formula for K l n (t m ; x) which will be used to calculate moments of the lth order Kantorovich-type Szász-Mirakjan operators.
where S n (f , x) is the Szász-Mirakjan operator defined in [23].

Now by direct calculation we write
Moments and central moments play an important role in approximation theory. In the following lemma we give explicit formulas for the mth (m = 0, 1, 2, 3, 4,) order moments of the lth order Kantorovich-type Szász-Mirakjan operators K l n (f ; x).
Lemma 3 For all n ∈ N and x ∈ [0, ∞), we have the following equalities: Proof The proof is done by using the recurrence formula given in Lemma 2.
K l n (t 3 ; x) and K l n (t 4 ; x) can be done in a similar way.
In the following lemma we give formulas for the mth order central moments of the lth order Kantorovich-type Szász-Mirakjan operators for m = 1, 2, 4.
Lemma 4 For all n ∈ N, we have the following central moments: Proof The proof is done by using Lemma 3 and the linearity of the operators.
One of the main problems in approximation theory is to estimate the rate of convergence for sequences of positive linear operators. Voronovskaja-type formulas are one of the most important tools for studying their asymptotic behavior. In the following lemma we give two limits that later will be used to prove Voronovskaja-type theorem for the lth order Kantorovich-type Szász-Mirakjan operators.
Lemma 5 For x ∈ [0, ∞) and n → ∞, we have the following limits: Proof The proof is trivial with the use of the formulas K l n (tx; x) and K l n ((tx) 2 ; x) given in Lemma 3,

Local approximation
In this section, we establish local approximation theorem for the lth order Kantorovichtype Szász-Mirakjan operators. We consider the Peetre's K -functional Then from the known result in [4], there exists an absolute constant C > 0 such that where is the second modulus of smoothness of f ∈ C B [0, ∞).
In the following theorem we state the first main result for the local approximation of our operators K l n (f ; x).

Theorem 6
There exists an absolute constant C > 0 such that . Note that K l n ((tx); x) = 0. By using Taylor's formula, we have Applying K l n to both sides of the above equation, we have On the other hand, We also have Using (2) and the uniform boundedness of K l n , we get If we take the infimum on the right hand side over all g ∈ C 2 B [0, ∞), we obtain which together with (1) gives the proof of the theorem.

Corollary 7 Let A > 0.
Then, for each f ∈ C[0, ∞), the sequence of operators K l n (f ; x) converges to f uniformly on [0, A].

Thus we get
By taking δ = √ α n (A), we get the desired result.
In the following theorem we give a Voronovskaja-type result for the lth order Kantorovich-type Szász-Mirakjan operators.
Theorem 10 For any f ∈ C 2 B [0, ∞), the following asymptotic equality holds: x ∈ [0, ∞) be fixed. By using Taylor's formula, we write where the function r(t, x) is the Peano form of the remainder, r(t, x) ∈ C B [0, ∞) and lim t→x r(t, x) = 0. Applying K l n to (3), we obtain By using the Cauchy-Schwarz inequality, we get We observe that r 2 (x, x) = 0 and r 2 (., x) ∈ C B [0, ∞). Now from Corollary 7 it follows that holds, then, for each x ∈ [0, ∞), we have where λ n (x) = 3l 2 +l 12(n+l) 2 + n-l 2 (n+l) 2 x + l 2 (n+l) 2 x 2 , L is a constant depending on α and f , and d(x, S) is the distance between x and S defined as Proof LetS be the closure of S in [0, ∞). Then there exists a point x 0 ∈S such that |xx 0 | = d(x, S). By the triangle inequality and by (6), we get Now, by using the Hölder inequality with p = 2 α and q = 2 2-α , we get and the proof is completed.

Weighted approximation
In this section, we give weighted approximation theorems for the lth order Kantorovichtype Szász-Mirakjan operators. We will use the following two lemmas which can be found in [3] and [12].

Lemma 13
Let m ∈ N ∪ {0} and l ∈ Z + be fixed. Then there exists a positive constant C m (l) such that Moreover, for every f ∈ C * 2 [0, ∞), we have Thus K l n is a linear positive operator from C * Proof Inequality (8) is obvious for m = 0. Let m ≥ 1. Then, by Lemma 12, we have where C m (l) is a positive constant depending on m and l. On the other hand, x m for every f ∈ C * m [0, ∞). By applying (8), we obtain (9).
Proof To prove this theorem, we need to use a Korovkin-type theorem on weighted approximation. That is, it is sufficient to verify the following three conditions: For m = 0, it is obvious. For m = 1, we have and by a similar way, we can write l 2 + 2nl (n + l) 2 + n(l + 1) (n + l) 2 + 3l 2 + l 12(n + l) 2 , which implies that Proof For any fixed 0 < A < ∞ and by Lemma 13, we have Using Theorem 9, we can see that J 1 goes to zero as n → ∞. By Theorem 14, we can get 1 + x m + x + x m+1 1 + x m+1 .
In the above inequality, if we substitute 1 √ n+l instead of δ, we obtain the desired result.

Conclusion
In this paper, by using the lth order integration and the definition of the Kantorovich type Szász-Mirakjan operators, we defined a new lth order Kantorovich-type Szász-Mirakjan operator. We derived a recurrence formula, and with the help of this formula we calculated the moments K l n (t m ; x) for m = 0, 1, 2, 3, 4 and we calculated the central moments K l n ((tx) m ; x) for m = 1, 2, 4. We established a local approximation theorem, a Korovkintype approximation theorem, and a Voronovskaja-type theorem. We obtained the rate of convergence of these types of operators for Lipschitz-type maximal functions, second order modulus of smoothness, and Peetre's K -functional. At last we investigated weighted approximation properties of the lth order Szász-Mirakjan-Kantorovich operators in terms of the modulus of continuity.