Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators.


Introduction
The integral modification of Baskakov operators having weight function of some beta basis function are defined as the following: for  ∈ [0, ∞),  > 0, () being the Dirac delta function.
The operators defined by (1) were introduced by Gupta [1]; these operators are different from the usual Baskakov-Durrmeyer operators.Actually these operators satisfy condition  , ( + , ) =  + , where  and  are constants.In [1], the author estimated some direct results in simultaneous approximation for these operators (1).In particular case  = 1, the operators (1) reduce to the operators studied in [2,3].
In recent years a lot of work has been done on such operators.We refer to some of the important papers on the recent development on similar type of operators [4][5][6][7][8][9].The rate of convergence for certain Durrmeyer type operators and the generalizations is one of the important areas of research in recent years.In present article, we extend the studies and here we estimate the rate of convergence for functions having derivative of bounded variation.
(ii) having a derivative   on the interval (0, ∞) coinciding a.e. with a function which is of bounded variation on every finite subinterval of (0, ∞).
It can be observed that all function  ∈   (0, ∞) possess for each  > 0 a representation
Remark 2. From Lemma 1, using Cauchy-Schwarz inequality, it follows that Lemma 3. Let  ∈ (0, ∞) and  , (, ) be the kernel defined in (1).Then for  being sufficiently large, one has Proof.First we prove (a); by using Lemma 1, we have The proof of (b) is similar; we omit the details.
Proof.By the application of mean value theorem, we have Also, using the identity where we can see that Also, Substitute value of   () from ( 12) in (11) and using ( 14) and ( 15), we get where  = /( − ).
On the other hand, we have Applying Holder's inequality, Remark 2, and Lemma 1, we have Also, Combining the estimates ( 17)-( 21), we get the desired results.This completes the proof of Theorem.

Rate of Convergence for Stancu Type Generalization of 𝐵 𝑛,𝛾
In 1968, Stancu introduces Bernstein-Stancu operators in [10], a sequence of the linear positive operators depending on two parameters  and  satisfying the condition 0 ≤  ≤ .
where the auxiliary functions   and ⋁   () were defined in Theorem 4.
The proof of the above theorem follows along the lines of Theorem 4; thus we omit the details.