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.
The integral modification of Baskakov operators having weight function of some beta basis function are defined as the following: for
The operators defined by (
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 [
We denote
By
having a derivative
It can be observed that all function
Let the function
Then it is easily verified that, for each
From the recurrence relation, it can be easily be verified that for all
From Lemma
Let
First we prove (a); by using Lemma
The proof of (b) is similar; we omit the details.
Let
By the application of mean value theorem, we have
Also, using the identity
we can see that
Also,
Substitute value of
Using Lemma
On applying Lemma
On the other hand, we have
Applying Holder’s inequality, Remark
Also,
Combining the estimates (
This completes the proof of Theorem.
In 1968, Stancu introduces Bernstein-Stancu operators in [
If we define the central moments, for every
then
For
From the recurrence relation, it can be easily verified that for all
Observe that
From Lemma
Applying the Cauchy-Schwarz inequality and keeping the same condition as in Remark
Let
The proof is the same as Lemma
Let
The proof of the above theorem follows along the lines of Theorem
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are thankful to the anonymous referee for making valuable comments leading to the better presentation of the paper. Special thanks are due to Professor Dr. Abdelalim A. Elsadany, Editor of Journal of Difference Equations, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.