A new construction of Lupaş operators and its approximation properties

The aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences um and vm of functions. We prove that the new operators provide better weighted uniform approximation over [0,∞). In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ . Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.


Introduction
The Weierstrass approximation theorem is the basis of approximation theory introduced by Weierstrass [15], which states that each continuous function defined on [a, b] can be approximated uniformly by some polynomial. In 1912, Bernstein [3] established a constructive proof of the Weierstrass theorem by using Korovkin's theorem [9].
On the other hand, Cárdenas et al. [4] defined the Bernstein-type operators by B m (goρ -1 )oρ and also presented a better degree of approximation depending on ρ. This type of approximation operators generalizes the Korovkin set from {e 0 , e 1 , e 2 } to {e 0 , ρ, ρ 2 }. In 2014, Aral et al. [1] also proposed a new modification of Szász-Mirakyan type operators to investigate approximation properties of the announced operators acting on functions defined on unbounded intervals [0, ∞). For various other generalizations of Szász-Mirakyan type operators, one can go through these papers [12][13][14] of Srivastava et al.
If we put ρ(z) = z in (1.1), then it reduces to the classical Lupaş operators defined in [11].
Very recently, a new construction of Szász-Mirakjan operators was given by Aral et al.
[2] by using ρ and two sequences of functions α m , β m defined on a subinterval of [0, ∞): Inspired by the idea used by Aral et al. in [2], in this paper we define a new construction of Lupaş operator (1.1) which depends on α m (z) and β m (z), where α m (z) and β m (z) are sequences of functions defined onẼ ⊂ [0, ∞).
The paper is organized as follows. In Sect. 2, the construction of the announced operator is presented and its moments and central moments are calculated. In Sect. 3, we study convergence properties by using weighted space. In Sect. 4, we obtain the rate of convergence of new constructed operators associated with the weighted modulus of continuity. In Sect. 5, we prove Voronovskaya-type asymptotic formula. Finally, in Sect. 6, we give some approximation results related to K-functional, also we define a Lipschitz-type functions.

The construction of Lupaş-type operators
Let g be a continuous functions where α m , β m are positive functions defined onẼ.
Then we consider the new operators in the following form: where ρ is a function which satisfies the conditions (ρ 1 ) and (ρ 2 ).
We will impose some assumptions on these operators, to show the sequence of operators (2.2) is an approximation process.
We suppose that, for z ∈Ẽ, where u m :Ẽ → R. From (2.2), we obtain Thus, we get Secondly, we assume that and where u m (z) > -1 for any z ∈Ẽ and m ∈ N 1 . Therefore, as a consequence, operators (2.2) become for m ∈ N 1 and for any z ∈Ẽ. We can recover some linear positive operators which are already in the literature. From operators (2.9) and for the particular choices of the functions u m , v m , and ρ: (i) If we take u m (z) = v m (z) = 0, operators (2.9) turn out to be operators (1.1).
(ii) If we take u m (z) = v m (z) = 0, ρ(z) = z, operators (2.9) turn out to be the classical Lupaş operators given in [11] by Now, in order to obtain weighted approximation processes, we assume that the following inequalities hold: From (2.10) and (2.11) it is clear that (L ρ m (g; z)) m≥m 0 is an approximation process onẼ ⊂ [0, ∞). Now, we give some lemmas which are required to prove our main results.

Lemma 2.1
For the operators L ρ m (g; z) and for all z ∈Ẽ, we have: 3 (1+u m (z)) 2 + 6(ρ(z)+v m (z)) 2 m(1+u m (z)) + 6 Lemma 2.2 By using Lemma 2.1 and by the linearity of operators L ρ m , we can acquire the central moments as follows:

Direct result in weighted space
In this section, by using weighted space, we discuss some convergence properties for the operators L ρ m . Let (z) = 1 + ρ 2 (z) be a weight function and B [0, ∞) be the weighted spaces defined as follows: In [6], Gadjiev proved the following results for the weighted Korovkin-type theorems.
Then, for any function g ∈ C * [0, ∞), we obtain Therefore, our result follows.  As we know each L ρ m (g; z) is defined onẼ. Now, by considering the following sequence of operators, we extend it on [0, ∞): Obviously, By applying 3.2 to the operators G m = A m the claim will be proved. Hence, we have to prove that By using (3.2), we get the desired result.

Rate of convergence
In this section, by using weighted modulus of continuity ω ρ (g; δ), we determine the rate of convergence for L ρ m which was recently considered by Holhoş [7] as follows: where g ∈ C [0, ∞), with the following properties: Proof If we calculate the sequences (a m ), (b m ), (c m ), and (d m ), then by using Lemma 2.1, clearly we have Finally, Thus conditions (4.1)-(4.5) are satisfied. Now, by Theorem 4.1, we obtain the desired result.

Voronovskaya-type theorem
In this section, we establish Voronovskaya-type result for L ρ m .

Local and global approximation
In order to prove local approximation theorems for the operators, let C B [0, ∞) be the space of real-valued continuous and bounded functions g with the norm · given by We begin by considering the K-functional where δ > 0 and W 2 = {s ∈ C B [0, ∞) : r , r ∈ C B [0, ∞)}. Then, in view of the known result [5], there exists an absolute constant D > 0 such that For g ∈ C B [0, ∞), the second order modulus of smoothness is defined as and the usual modulus of continuity is defined as Theorem 6.1 There exists an absolute constant D > 0 such that where g ∈ C B [0, ∞) and Proof Let r ∈ W 2 and z, ζ ∈ [0, ∞). By using Taylor's formula we have By using the equality putting v = ρ(z) in the last term in equality (6.2), we get By applying S * μ,λ m,ρ to (6.2) and also by using Lemma 2.1 and (6.4), we deduce By using conditions (ρ 1 ) and (ρ 2 ), we get Hence we have Taking infimum over all r ∈ W 2 , we obtain Let 0 < α ≤ 1, ρ be a function with conditions (ρ 1 ), (ρ 2 ) and Lip M (ρ(y); α), H ≥ 0 satisfying where H α,g is a constant.
By applying Hölder's inequality with p = 2 α and q = 2 2-α , we have which proves the desired result.
Conclusion. Here, a new construction of the generalized Lupaş operators is constructed. We have investigated convergence properties, order of approximation, Voronovskaja-type results, and quantitative estimates for the local approximation. The constructed operators have better flexibility and rate of convergence which depend on the selection of the function ρ and the sequences u m , v m .