Rate of convergence by Kantorovich-Szász type operators based on Brenke type polynomials

The present paper deals with the approximation properties of the univariate operators which are the generalization of the Kantorovich-Szász type operators involving Brenke type polynomials. We investigate the order of convergence by using Peetre’s K-functional and the Ditzian-Totik modulus of smoothness and study the degree of approximation of the univariate operators for continuous functions in a Lipschitz space, a Lipschitz type maximal function and a weighted space. The rate of approximation of functions having derivatives equivalent with a function of bounded variation is also obtained.


Introduction
Linear positive operators play an important role in the study of approximation theory. One of the best known among these operators is the Szász operator [] S n (f ; x) = e -nx [] considered a Stancu type modification of the Szász Kantorovich operators involving Brenke type polynomials and obtained the degree of approximation by means of the classical modulus of continuity and Peetre's K-functional. A Voronovskaya type theorem for the considered operators including Gould-Hopper polynomials was also proven. Mursaleen and Ansari [] presented a Chlodowsky type generalization of Szász operators defined by using Brenke type polynomials and studied the order of convergence for functions in a weighted space besides other classical approximation results. Recently where {α n }, {β n } are strictly increasing sequences of positive numbers such that and p k (x) = ∞ r= a k-r b r x r , k = , , , . . . , are Brenke type polynomials having the following generating function: where A(t), B(t) are analytic functions such that () and the power series () converges for |t| < R (R > ), (iv) lim y→∞ B (k) (y) B(y) =  for k = , , , . Atakut and Büyükyazici [] studied the order of convergence of the operators defined in (). For the special case α n = β n = n, the operators () are reduced to the Kantorovich variant of a generalization of Szász operators based on the Brenke type polynomials. For some other significant papers dealing with the generalization of Szász type operators, we refer to [-].
In the present paper we establish the order of approximation of the operators () by using Peetre's K-functional and the Ditzian-Totik modulus of smoothness. Also, we study the rate of convergence of these operators for a Lipschitz space, a Lipschitz type maximal function and a weighted space. The rate of approximation of functions having derivatives equivalent to a function of bounded variation is also obtained.

Preliminaries
To examine the approximation properties of the operator L α n ,β n n defined in (), we give some basic results in the form of lemmas as follows.
As a consequence of Lemma ., we have the following.

Local theorem of approximation
In what follows, let L Further, M denotes a constant not necessarily the same at each occurrence.
The Lipschitz type space [] is defined as follows: for some M f >  and  < r ≤ . Several researchers have considered the approximation of functions in this space for different sequences of linear positive operators (cf. [, -] etc.).
Operating by L α n ,β n n on the above equation, we get Now, applying Hölder's inequality with p = /r, q = /(r) and using Lemma ., we get where x ∈ [, ∞) and r ∈ (, ]. For contributions of researchers on approximation of functions in this space, we refer to [, , , ]. Proof By equation (), Applying the operator L α n ,β n n on both sides of the above equation, and then using Lemma . and Hölder's inequality with p = /r, q = /(r), we get Thus, we get the desired result.
For f ∈C B [, ∞) and δ > , the first order modulus of continuity is defined as and Peetre's K-functional is given by holds for all δ >  and where ω  is the second order modulus of smoothness of f ∈ C B [, ∞), which is defined as follows: .
Proof For f ∈C B [, ∞), we define the auxiliary operator as follows: After taking the modulus of both sides and applying Lemma ., we obtain Now, by Lemma ., we get Let g ∈C  B [, ∞), then using Taylor's theorem we can write OperatingL α n ,β n n on the above equation and using (), we get Now taking the modulus on both sides, we obtain Therefore, by using g C  Now, using the definition of auxiliary operators (), we have

Combining (), () and () with the above equation, we get
and, after taking the infimum on the right-hand side over all g ∈C  B , we obtain This completes the proof.
Now, using the well-known property of modulus of continuity, for δ >  and f ∈C  Therefore, After applying the Cauchy-Schwarz inequality, we have Choosing δ =   δ α n ,β n (x), we get our result.
, the Ditzian-Totik modulus of smoothness of first order is given by and appropriate Peetre's K-functional is defined by Now, we find the order of approximation of the operator () by means of the Ditzian-Totik modulus of smoothness.
Theorem . For any f ∈C B [, ∞) and x ∈ (, ∞), Proof By Taylor's theorem, for any g ∈ W φ [, ∞), we get therefore, which gives For any g ∈ W φ [, ∞), using Lemma . and the above equation, we get After applying the Cauchy-Schwarz inequality, we obtain Taking the infimum on the right-hand side over all g ∈ W φ [, ∞), we get which leads us to the required result with the help of the relation between Peetre's Kfunctional and the Ditzian-Totik modulus of smoothness as given in ().

Weighted approximation
. Proof Let x  >  be arbitrary but fixed, then in view of () we get By the Korovkin theorem, the sequence {L α n ,β n n (f ; x)} converges uniformly to the function f on every closed interval [, a], as n → ∞, (cf. [], p.), therefore for a given > , ∃n  ∈ N such that By using Lemma ., we can find n  ∈ N such that Hence Now, using (), we get Let n  = max{n  , n  }, then by (), () and () Choose x  to be so large that Then, combining (), () and (), we have Hence the proof is done.
where g denotes a function of bounded variation on every finite subinterval of [, ∞).
In order to study the convergence of the operators L α n ,β n n for the functions having a derivative of bounded variation, we rewrite the operator () as follows: where W (t, x) is the kernel given by .
Proof By hypothesis (), where Now, using Lemma ., equations () and (), we get Since t x δ x (u) du = , Now, we break the second term on the right-hand side of the above equation as follows: x t x f x (u) du W (t, x) dt = -I  + I  , where I  = x  x t f x (u) du W (t, x) dt, Taking the modulus on both sides of (), we get After applying the Cauchy-Schwarz inequality, we get Now applying Lemma . and integration by parts, I  can be written as After taking the modulus, we have x-x/ √ n f x (t) λ α n ,β n (t, x) dt = K  + K  , say.

Conclusion
We study the order of approximation of the Kantorovich-Szász type operators based on Brenke type polynomials with the aid of Peetre's K-functional and the Ditzian-Totik modulus of smoothness. The rate of convergence of these operators for functions in a Lipschitz type space and a weighted space is investigated. The degree of approximation of functions whose derivatives coincide a.e. with a function of bounded variation is also discussed.