Voronovskaja-Type Quantitative Results for Differences of Positive Linear Operators

: We consider positive linear operators having the same fundamental functions and different functionals in front of them. For differences involving such operators, we obtain Voronovskaja-type quantitative results. Applications illustrating the theoretical aspects are presented.


Introduction
The problem of studying the differences of positive linear operators was formulated firstly by Lupaş [1]. In particular, he was interested in the commutator (A, B) := AB − BA, due to its property of antisymmetry. Generally speaking, there are two approaches to estimate the difference of two positive linear operators. In this context, there are two approaches to estimate the difference of two operators. One of them deals with operators that have the same moments up to a certain order. For detailed historical background, we refer to the work of Acu et al. [2] and the references therein. The other approach considers those operators that have the same fundamental functions and different functionals in their construction (see [2,3]). In the second perspective, the discrete operator associated with an integral operator has important role in the study. Raşa [4] noticed the advantages of the discrete operators associated with certain integral operators in this area. In this sense, it is helpful to mention the work of Heilmann et al. [5] from which the notion of discrete operator is reproduced below [2]: Let I ⊂ R denote an interval and H be a subspace of C(I) containing the monomials e i (x) = x i , i = 0, 1, 2. Consider a positive linear operator L : H → C(I) satisfying Le 0 = e 0 given by where F k : H → R are positive linear functionals satisfying F k (e 0 ) = 1, and p k ∈ C(I) are the fundamental functions such that p k ≥ 0 and ∞ ∑ k=0 p k = e 0 . The discrete operator associated with L is denoted by D and defined as where b F k := F k (e 1 ). Namely, the functional in the construction of the discrete operator is the point evaluation at b k , which is obviously simpler than the functional F k of the corresponding operator L. Therefore, it is easier to work with the discrete operator associated with L. In [3], some useful estimates for the differences of certain positive linear operators with the same fundamental functions were studied.
In the present note, we study the difference of positive linear operators, with the same fundamental functions, by obtaining Voronovskaja-type quantitative estimates.

Preliminaries
Throughout the paper, we shall adopt the same notation of [3]. Thus, E(I) will denote a space of real valued and continuous functions defined on I containing the polynomials, and E b (I) will denote the space of all functions f from E(I) having For a positive linear functional F satisfying F(e 0 ) = 1, the following expressions will be used: Obviously, we have Moreover, for convenience, we adopt the notation Thus, since the functional F is linear, one has Recall that the remainder R 2 f ; b F , . of Taylor's formula is given by where ξ is between x and b F . Therefore, since µ F 1 = 0, one has Using the fact that |ξ − Here Thus, for δ > 0, it follows that Therefore,

Main Result
As in [3], let K denote a set of non-negative integers and denote fundamental functions satisfying ∑ k∈K p k = e 0 . Let F k and G k be two positive linear . Now, we deal with the positive linear operators U and V, acting from D(I) to C(I), given by Let D U and D V denote the discrete operators associated with U and V, which are given by respectively. For future correspondence, we denote and Moreover, from (2), the ith central moment of each operator can be written as In [3], the authors measured the distance |U( f ; x) − V( f ; x)| using properties of the associated discrete operators. Specifically, they obtained the following result: where σ(x) is given by (8) and t := sup [3]

[Theorem 3]).
A natural question arising here is to estimate the difference of positive linear operators in the sense of Videnskiȋ who stated the well-known result of Voronovskaja [6] for the Bernstein operators in the following quantitative form.
where ω( f , .) is the modulus of continuity of f .
In this context, we give an expression for the difference of a positive linear operator and its discrete operator.

Lemma 1. Let x be an arbitrary point in I and f
where R 2 f ; b F , · is the remainder of Taylor's formula given by (4).
Proof. Let x ∈ I be a given point. Then, from (5), it readily follows that Now we present a quantitative Voronovskaja-type theorem for the difference U − V.
Proof. Let x be an arbitrary fixed point in I. Using (5), we obtain The above formula can be expressed as By using (3), the last formula can be written as The term in (11) is the remainder R 2 f ; x, b F of Taylor's formula for x (fixed) and b F , given by where ξ is a point between x and b F . Therefore, we have The formula (11) can be written as Taking into account (6), (7), and (12), we obtain we obtain Using (8) and (9), the theorem is proved.

Quantitative Voronovskaja-Type Result for the Differences of Bernstein Operators and Kantorovich Operators
The well-known Bernstein operators B n : C[0, 1] → C[0, 1], n ∈ N are given by where the fundamental functions p n,k (x) are The Kantorovich operators K n : L 1 [0, 1] → C[0, 1], n ∈ N are defined as ( [8]) In [3] [Proposition 8], as an application of Theorem 1, the authors expressed the difference between Bernstein and its Kantorovich variant as Now, we give an estimate of this difference with the help of Theorem 3.

Quantitative Voronovskaja-Type Result for the Differences of Bernstein Operators and Genuine Bernstein-Durrmeyer Operators
For f ∈ C[0, 1], the genuine Bernstein-Durrmeyer operators are defined as In [3] [Proposition 4], as an application of Theorem 1, the authors expressed the difference between Bernstein and Bernstein-Durrmeyer operators as Below, we estimate this difference via the above quantitative Voronovskaja-type result.
From (3), we obtain From this point of view, the argument of the modulus of continuity appearing in Theorem 3 is easier to evaluate than (13), since it is represented in terms of the point evaluation functional of the corresponding discrete operators, whereas in (13), the forth central moment of each operator must be calculated.
Differences of other pairs of operators will be considered in a forthcoming paper. The special case of the commutator AB-BA will be considered, due to its property of antisymmetry.
Author Contributions: The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.