On boundedness inequalities in the variation of certain Schurer-type operators

This paper is concerned with boundedness inequalities in the variation for the higher order derivatives of general Schurertype operators. In particular, the boundedness inequalities in the variation for the higher order derivatives of the Bernstein–Schurer, Kantorovich–Schurer, and Durrmeyer–Schurer operators are derived.


INTRODUCTION
The Bernstein polynomials, have influenced many branches of the approximation theory and their properties have been prototypes for our research as well.
Dealing with the class of functions of bounded variation BV [0, 1], the Bernstein polynomials have the total variation diminishing property where V [0,1] [ f ] is the total variation of f and f ∈ BV [0, 1] (see [10], which is the first paper in this direction).
In [2] this has been called the variation detracting property (VDP).The total variation diminishing property of this kind is known for many positive operators.In the case of the Kantorovich operators where the variation detracting property holds as follows.
For the Bernstein operator there has been some interest in investigating the variation detracting property for the derivatives in the form (see, e.g.[9]) (1.4) In this paper we investigate boundedness inequalities for the higher order derivatives of some, quite general, Schurer-type operators, in particular cases the Bernstein-Schurer, the Kantorovich-Schurer, and the Durrmeyer-Schurer operators.
Let L n be a polynomial positive operator, i.e., we have a polynomial Due to (1.1) and (1.4) there arises the question of determining the constant M r > 0, independent of f , f (r) ∈ BV [0, b], for which We call inequality (1.5) the boundedness inequality.If M r ≤ 1, we call inequality (1.5) the variation detracting property.

BOUNDEDNESS INEQUALITIES FOR DERIVATIVES OF SCHURER-TYPE OPERATORS
In [1,4,11,12], certain Schurer-type operators are defined and their approximation properties are investigated.We investigate boundedness inequalities (1.5) of some general Schurer-type operators in a unified approach.Let where F k,n,p ( f ) is some positive linear functional of f ∈ C[0, 1 + p], p = 0, 1, 2, ..., and We consider here the following cases: 1.If in (2.1) we put a = 1 and then we get the Bernstein-Schurer operator B n,p ; the subcase p = 0 gives us the Bernstein operator B n .
2. If in (2.1) we put a = 1 and then we get the Kantorovich-Schurer operator K n,p ; the subcase p = 0 gives us the Kantorovich operator K n .3. If in (2.1) we put a = p + 1 and then we get the Durrmeyer-Schurer operator D n,p ; the subcase p = 0 gives us the Durrmeyer operator D n .
Since U n,p,a f as a polynomial is continuously differentiable on [0, a], then for the left-hand side of (1.5) it is known that So let us find the r + 1-th derivative of the polynomial U n,p,a f .We introduce differences by the first index: We use next a lemma, which is generalized from [14], Chap.II, §19, Lemma 2 (see also [6], p. 306, formula (2.3)).
The following proposition gives us a general idea for studying the variation detracting property for operators (2.1).
Proof.Since the beta function yields by Lemma 1 we have Next, we have to express differences through derivatives.First, we need a definition of r − 1 times absolutely continuous functions on the interval [0, b] (compare, e.g.[3], p. 7).Definition 1.We say that f ∈ AC r−1 [0, b], r ∈ N, the space of all (r − 1)-times absolutely continuous functions on We introduce differences The class AC r−1 [0, b] allows us to represent the differences △ r h f (x) via the derivatives The next lemma is from [13] (see Chap. 3, §3, formula (4)).
Now we have to calculate, according to (2.7), the sum of differences (2.9). ) Proof.By Lemma 2 we have Taking the sum we get By introducing the new variables ) and the Jacobian determinant J = 1.We get from (2.12) and (2.13) the estimate Hence, the integral on the right-hand side of inequality (2.11) is estimated by From (2.11) and (2.14) we obtain our assertion.
Let us first investigate the boundedness inequality for the derivatives of the Bernstein-Schurer polynomials. (2.15) Proof.For the Bernstein-Schurer polynomials In the case r = 0 the proof is almost identical to the proof of Proposition 3.1 in [2].In the case r = 1, ..., n + p − 1 by Proposition 1 and Lemma 3 (v = 0) we have As a corollary, we get now a statement for the Bernstein operators, proved also in [14], Chap.II, §19, Lemma 3.
Corollary 1.Let f ∈ AC r [0, 1], r = 0, 1, ..., n−1.Then for the arbitrary derivatives of the Bernstein operator the VDP holds, i.e. (2.16) For the proof we take in Theorem 1 p = 0. Similarly to the case of derivatives of the Bernstein-Schurer polynomials we can investigate the boundedness inequality for the derivatives of the Kantorovich-Schurer polynomials. (2.17) Proof.In the case r = 0 the proof is almost identical to the proof of Proposition 3.3 in [2].In the case r = 1, ..., n + p − 1 by Proposition 1 (a = 1) and Lemma 3 we have )|du.
Taking s = u + v n+1 , t = u we have We get now in the case p = 0 Then for the arbitrary derivatives of the Kantorovich operator the VDP holds, i.e.
To investigate the variation detracting property for the Durrmeyer-Schurer operators by Proposition 1 we need to calculate △ r+1 F k,n,p ( f ).By definition (2.5) we write It appears that the differences of the basic polynomials p k,n+p,p+1 (t) can be represented via derivatives.The next result in a particular case is obtained in [5], proof of Theorem II.6, p. 332; however, for the completeness of the presentation we will give an elementary proof.where m ≥ k ≥ r ≥ 0, x ∈ [0, a], holds.
Proof.We prove (2.18) by induction on r.For r = 0 it is obvious by definition.For r = 1 we get

.19)
Let us assume that (2.18) holds for some 1 ≤ r < m.Differentiating (2.19) r times and using (2.18) we get To estimate the constant M r in (1.5) for the Durrmeyer-Schurer operators we need   (r) ].
In the case p = 0 we get the known result of Derriennic [5] (see the proof of Proposition 4.1) for the Durrmeyer operators.

CONCLUSIONS
We investigated the boundedness inequalities for the higher order derivatives of some general Schurer-type operators in a unified approach.In particular, we proved the boundedness inequality for the higher order derivatives of the Bernstein-Schurer, Kantorovich-Schurer, and Durrmeyer-Schurer operators.Moreover, we proved the variation detracting property for the arbitrary derivatives of the Bernstein, Kantorovich, and Durrmeyer-Schurer operators.For the arbitrary derivatives of the Bernstein-Schurer and Kantorovich-Schurer operators only the boundedness inequality with the constant M r on the right-hand side of the inequality that exceeds 1, holds.