The purpose of the present paper is to introduce the generalized form of Srivastava-Gupta operators and study their approximation properties.
We use analytical method to obtain our results.
We have established the rate of convergence, in simultaneous approximation, for functions having derivatives of bounded variation.
The results proposed here are new and have a better rate of convergence.
Keywords:Bounded variation; Srivastava-Gupta operators; Simultaneous approximation; Rate of convergence; 2010; 26A45; 41A28
In the year 2003, Srivastava and Gupta  introduced a general family of summation-integral type operators which includes some well-known operators as special cases. They estimated the rate of convergence for functions of bounded variation. For the details of special cases in , we refer the readers to [2-7]. Ispir and Yuksel  considered the Bezier variant of the operators studied in  and estimated the rate of convergence for functions of bounded variation. Very recently, Deo  studied Srivastava-Gupta operators and obtained the faster rate convergence as well as Voronovskaja type results for these operators by using the King approach. In the last section, he considered Stancu variant of these operators and established some approximation properties.
The operators Gn,c is defined as follows:
(iv) there exists an integer c such that
Nowadays, the rate of convergence for the functions having the derivatives of bounded variation (BV) is an interesting area of research. Bai et al. worked in this direction and estimated the rate of convergence for the several operators. Gupta  estimated the rate of convergence for functions of BV on certain Baskakov-Durrmeyer type operators. Ispir et al.  estimated the rate of convergence for the Kantorovich type operators for functions having derivatives of BV. Recently, Acar et al.  introduced the general integral modification of the Szász-Mirakyan operators having the weight functions of Baskakov basis functions. The rate of convergence for functions having the derivatives of bounded variation is obtained. This motivated us to study the rate of convergence for the generalized Srivastava-Gupta operators as follows: For a function the class of bounded variation functions satisfying the growth condition |f (t )|≤M (1 + t)αM>0, α≥0, the operators Gn,r,c are defined by
where pn,k(xc) is given by Equation 1.2 and n>(r−1)c.
For the special case of c=1, the operators in Equation 1.3 are reduced to the following operators:
We denote that the class of absolutely continuous functions f on (0,∞) by DBq(0,∞), (where q is some positive integer) are satisfied:
(ii) the function f has the first derivative on interval (0,∞) which coincide, a.e., with a function which is of bounded variation on every finite subinterval of (0,∞). It can be observed that for all f∈DBq(0,∞), we can have the representation
In the present paper, we study the rate of convergence for the operators Gn,r,c for functions having the derivatives of bounded variation. We also mention a corollary which provides the result in simultaneous approximation.
The principal methods used in the present work involve the application of the theory of functions having the derivatives of bounded variation to analyze and study the rate of convergence, in simultaneous approximation, for the Srivastava-Gupta operators.
Results and discussion
In the sequel we shall need the following lemmas:
If we define the moments as
Furthermore, Tn,r,m(x,c) is polynomial of degree m in x and
To compute I2 we have
Again using t(t−x)m=(t−x)m + 1 + x(t−x)m and integrating by parts, we get
Proceeding in a similar manner, we obtain J2 as
Combining I1,I2,J1, and J2, we have
Let x∈(0,∞) and λ>2; then for n sufficiently large, Lemma 1 yields that
Let x∈(0,∞) and λ>2; then for n sufficiently large, we have
The proof of the lemma follows easily by Remark 2. For instance, for the first inequality for n sufficiently large and 0≤y<x, we have
The proof of the second inequality follows along the similar lines. □
We prove this result by applying the principle of mathematical induction and using the following identity:
The above identities are true even for the case of k=0, as we observe that pn + c,k=0 for k<0. Using Equation 2.1 and integrating by parts, we have
which shows that the result holds for s=1. Let us suppose that the result holds for s=m i.e.,
Now by Equation 2.1,
Integrating by parts the last integral, we have
Thus, the result is true for s=m + 1; hence, by mathematical induction the proof of the lemma is completed. □
In this subsection we prove our main results.
Using the mean value theorem, we have
Also it is a valid identity that
Obviously, we have
Thus, using the above identities, we can write
Also it can be verified that
Combining Equations 3.1–3.3, we get
Applying Remark 2 and Lemma 1 in Equation 3.4, we have
In order to complete the proof of the theorem, it suffices to estimate the terms An,r(f,x) and Bn,r(f,x) as follows:
By using the Hölder’s inequality and Remark 2, we get the estimate as follows:
Collecting the estimates from Equations 3.6–3.9, we obtain
Through combining the Equations 3.4, 3.10, and 3.11, we get the desired results. □
As a consequence of Lemma 3, we can easily prove the following corollary for the derivatives of the operators Gn,r,c.
Let fs∈DBq(0,∞), q>0 and x∈(0,∞). The for λ>2 and n sufficiently large, we have
We have obtained the rate of convergence for the generalized Srivastava-Gupta operators for the functions having the derivatives of bounded variation which gives a better rate of convergence than the classical Srivastava-Gupta operators.
The authors declare that they have no competing interests.
DKV and PNA contributed equally to this work. Both authors read and approved the final manuscript.
DKV is a research fellow at the Department of Mathematics, Indian Institute of Technology Roorkee in Roorkee, India. PNA is a professor at the Department of Mathematics, Indian Institute of Technology Roorkee in Roorkee, India.
The authors are thankful to the referees for valuable suggestions, leading to an overall improvement in the paper. The first author is also thankful to the Ministry of Human Resource and Development India for the financial support to carry out the above work.
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