ON CONVERGENCE OF CHLODOVSKY TYPE DURRMEYER POLYNOMIALS IN VARIATION SEMINORM

. This paper deals with the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm.


INTRODUCTION
Let X loc [0, ∞) be the class of all complex-valued functions locally bounded on [0, ∞). For x ∈ X loc [0, ∞), the Chlodovsky polynomials C n f are defined as: (1) (C n f ) (x) = where n ∈ N and (b n ) is an increasing sequence of positive numbers satisfying lim n→∞ b n = ∞ and lim n→∞ bn n = 0. These polynomials were introduced by I. Chlodovsky [1] in 1937 in generalization of the Bernstein polynomials, the case b n = 1, n ∈ N, which approximate the function f on the interval [0, 1]. Some other generalizations of the Bernstein polynomials defined on unbounded sets can be found in [2], [3]. Works on Chlodovsky polynomials are fewer, since they are defined on an unbounded interval [0, ∞).
This generalizes Chlodovsky polynomials by incorporating Durrmeyer operators [4], hence the name Chlodovsky-Durrmeyer operators We may also mention that some articles related to Chlodovsky-Durrmeyer operators and their different generalizations are given in [5]- [6].
The main motivation for this paper is to study the variation detracting property and rate of approximation of the Chlodovsky type Durrmeyer polynomials in the space of functions of bounded variation with respect to the variation seminorm. The first research devoted to the variation detracting property and the convergence in variation of a sequence of linear positive operators was due to Lorentz [7]. Later in [8], authors have introduced, developed in details and studied the deep interconnections between variation detracting property and the convergence in variation for Bernstein-type polynomials and singular convolution integrals. After this fundamental study, the convergence in variation seminorm has become a new research field in the theory of approximation. For further reading on different operators, we refer to readers to [9]- [15].

NOTATION AND AUXILIARY RESULTS
For the notation; let I ⊂ R be a bounded or unbounded interval. We denote by V [I]  Some interesting properties of the space T V (I) are presented in [8].
In order to obtain a convergence result in the variation seminorm, it is necessary and important to state the variation detracting property. Let L be a linear operator acting on a given space S of real-valued functions defined on I such that BV (I) ⊂ S. The operator L possesses the variation detracting property if sequence (g n ) n≥1 , g n ∈ AC (I), n ∈ N, then also f ∈ AC [0, 1] and So, convergence in variation of (g n ) n≥1 ⊂ AC (I) to f, represents the convergence of the derivatives (g n ) n≥1 to f in the norm L 1 (I), the Banach space of all real-valued Lebesgue integrable functions defined on I. Let us define the sum moments as in [11]: where m ∈ N 0 (the set of non-negative integers). Then there hold the following identities (see, e.g., [11]) where A m (x) denotes a polynomial in x, of degree [m/2]−1, with non-negative coefficients independent of n, and [a] denotes the integral part of a. For the proof see Butzer-Karsli [16]. Since x bn , we can write the following representations for the first derivative of (D n f ) (x);

VARIATION DETRACTING PROPERTY OF CHLODOVSKY-DURRMEYER OPERATORS
In this section, we prove the variation detracting properties of the Chlodovsky-Durrmeyer Operators.
Proof. For convenience we write the Chlodovsky-Durrmeyer operators as: As in (7), differentiating (2) and putting ∆F k, Considering the representation (10) of (D n f ) , one has and so we get From (11) and (12), we obtain The desired estimate (8)  Thus, the proof of the theorem is complete.

RATE OF APPROXIMATION IN T V -NORM
This section deals with the rates of approximation D n g to g in the variation seminorm.
In order to obtain a convergence result in variation seminorm, we assume that lim Proof. By Taylor's formula with integral remainder term, one has From (13) Calculating (3) and (4), we obtain As in the proof of [15], we choose δ a sufficiently small positive real number, let's divide (R n g) (x) in to two parts as follows; In order to estimate the integration domain of the double integral in the remainder term (14), we divide the summation into different sums as following; (R n,1 g) (x) = A 2,n g + A 5,n g and (R n,2 g) (x) = A 1,n g + A 3,n g + A 4,n g + A 6,n g.
Now we only estimate A i,n g for i = 1, 2, 5, and 6 respectively. Firstly, let us estimate A 1,n g as follows; we get Analogously, A 2,n g can be estimated by In view of Hölder inequality, (4) and (5), we get As to the term A 5,n g, noting (4) and (5)