On Rate of Approximation by Modified Beta Operators
We establish the rate of convergence for the modified Beta operators , for functions having derivatives of bounded variation.
In the year 1995 Gupta and Ahmad  proposed modified Beta operators so as to approximate Lebesgue integrable functions on as where These operators are linear positive operators and reproduce only the constant functions. We set In particular, it can be observed by the definition of Beta function that
Gupta and Ahmad  estimated asymptotic formula and error estimate for these operators in simultaneous approximation. Later Gupta et al.  studied the approximation properties of these operators in norm and they obtained some direct results for the linear combinations. Here we continue our studies on these operators. First we consider the following class of functions.
By we mean the class of absolutely continuous functions defined on , which satisfy the following:(i),(ii)this class has a derivative on the interval coinciding a.e. with a function which is of bounded variation on every finite subinterval of It can be observed that all functions possess for each the representation
Very recently Gupta and Agrawal  and Gupta et al.  have obtained an interesting result on the rate of convergence of certain Durrmeyer type operators in ordinary and simultaneous approximation. spir et al.  estimated similar results for Kantorovich operators for functions with derivatives of bounded variation. We now extend the study for the modified Beta operators and estimate the rate of convergence of the operators (1) for functions having derivatives of bounded variation.
2. Auxiliary Results
We shall use the following lemmas to prove our main theorem.
Lemma 2.1 (). Let the function , , be defined as Then Also for each , one has
Remark 2.2. From Lemma 2.1 and using Cauchy-Schwarz inequality, for sufficiently large, it follows that where .
Lemma 2.3. Let , then for and sufficiently large, one has
Proof. By using Lemma 2.1, for sufficiently large, we have The proof of the second inequality is similar, and we skip the details.
3. Rate of Approximation
Our main result is stated as follows.
Theorem 3.1. Let ,??, and . Then for sufficiently large, one has where the auxiliary function is given by denotes the total variation of on
Proof. By mean value theorem, we have
Applying the identity
Equation (3.3) becomes (keeping in mind that the last term of the above identity vanishes)
Thus in view of the above values, Lemma 2.1, and Remark 2.2, (3.5) reduces to
In order to complete the proof of the theorem, it is sufficient to estimate the terms and Applying integration by parts, using Lemma 2.3 and taking , we have
Let Then we have
On the other hand, we have
For the estimation of the first two terms in the right-hand side of (3.11).we proceed as follows.
Applying Holder's inequality, Remark 2.2, and Lemma 2.1, we have Also, by Remark 2.2, the third term of the right side of (3.11) is given by Thus Combining the estimates (3.7), (3.10), and (3.14) we get the desired result. This completes the proof of the theorem.