Abstract
Purpose
The purpose of the present paper is to introduce the generalized form of SrivastavaGupta operators and study their approximation properties.
Methods
We use analytical method to obtain our results.
Results
We have established the rate of convergence, in simultaneous approximation, for functions having derivatives of bounded variation.
Conclusions
The results proposed here are new and have a better rate of convergence.
Keywords:
Bounded variation; SrivastavaGupta operators; Simultaneous approximation; Rate of convergence; 2010; 26A45; 41A28Introduction
In the year 2003, Srivastava and Gupta [1] introduced a general family of summationintegral type operators which includes some wellknown operators as special cases. They estimated the rate of convergence for functions of bounded variation. For the details of special cases in [1], we refer the readers to [27]. Ispir and Yuksel [8] considered the Bezier variant of the operators studied in [1] and estimated the rate of convergence for functions of bounded variation. Very recently, Deo [9] studied SrivastavaGupta 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 G_{n,c} is defined as follows:
where
and
Here is a sequence of functions defined on the closed interval [0,b, b>0, satisfying the following properties. For each and :
(ii) ϕ_{n,c}(0)=1,
(iii) ϕ_{n,c} is completely monotone so that and
(iv) there exists an integer c such that
(see [1]).
Nowadays, the rate of convergence for the functions having the derivatives of bounded variation (BV) is an interesting area of research. Bai et al.[10] worked in this direction and estimated the rate of convergence for the several operators. Gupta [4] estimated the rate of convergence for functions of BV on certain BaskakovDurrmeyer type operators. Ispir et al. [11] estimated the rate of convergence for the Kantorovich type operators for functions having derivatives of BV. Recently, Acar et al. [12] introduced the general integral modification of the SzászMirakyan 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 SrivastavaGupta operators as follows: For a function the class of bounded variation functions satisfying the growth condition f (t )≤M (1 + t)^{α}M>0, α≥0, the operators G_{n,r,c} are defined by
where p_{n,k}(xc) is given by Equation 1.2 and n>(r−1)c.
Remark 1
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 DB_{q}(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∈DB_{q}(0,∞), we can have the representation
In the present paper, we study the rate of convergence for the operators G_{n,r,c} for functions having the derivatives of bounded variation. We also mention a corollary which provides the result in simultaneous approximation.
Methods
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 SrivastavaGupta operators.
Results and discussion
In the sequel we shall need the following lemmas:
Lemma 1
If we define the moments as
and then, T_{r,n,0}(x,c)=1, and for n>(m + r + 1)c, we have the following recurrence relation:[n−(m + r + 1)c]T_{n,r,m + 1}(x,c)
Consequently,
Furthermore, T_{n,r,m}(x,c) is polynomial of degree m in x and
Proof
Taking the derivative of T_{n,r,m}(x,c) with respect to x and using the identity , we have
To compute I_{2} we have
Using p_{n−(r−1)c,k}(t,c), we can write I_{1} as
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 J_{2} as
Combining I_{1},I_{2},J_{1}, and J_{2}, we have
□
Remark 2
Let x∈(0,∞) and λ>2; then for n sufficiently large, Lemma 1 yields that
Lemma 2
Let x∈(0,∞) and λ>2; then for n sufficiently large, we have
Proof
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. □
Lemma 3
Let f be s times differentiable on [0,∞) such that f^{(s−1)}(t)=O(t^{q}) as t→∞ where q is a positive integer. Then for any and n>max{q,r + s + 1}, we have
Proof
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 p_{n + 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
Therefore,
Thus, the result is true for s=m + 1; hence, by mathematical induction the proof of the lemma is completed. □
Main results
In this subsection we prove our main results.
Theorem 1
Let f∈DB_{q}(0,∞), q>0 and x∈(0,∞). The for λ>2 and n sufficiently large, we have
where denotes the total variation of f_{x} on [a,b], and the auxiliary function f_{x} is defined by
Proof
Using the mean value theorem, we have
Also it is a valid identity that
where
Obviously, we have
Thus, using the above identities, we can write
Also it can be verified that
and
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 A_{n,r}(f,x) and B_{n,r}(f,x) as follows:
For estimating the integral above, we proceed as follows: since t≥2x implies that t≤2(t−x) Schwarz inequality which follows from Lemma 1,
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
On the other hand, to estimate B_{n,r}(f,x) by applying the Lemma 2 with and integration by parts, we have
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 G_{n,r,c}.
Corollary 1
Let f^{s}∈DB_{q}(0,∞), q>0 and x∈(0,∞). The for λ>2 and n sufficiently large, we have
where denotes the total variation of f_{x} on [a,b], and the auxiliary function D^{s + 1}f_{x} is defined by
Conclusions
We have obtained the rate of convergence for the generalized SrivastavaGupta operators for the functions having the derivatives of bounded variation which gives a better rate of convergence than the classical SrivastavaGupta operators.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DKV and PNA contributed equally to this work. Both authors read and approved the final manuscript.
Authors’ information
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.
Acknowledgements
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.
References

Srivastava, HM, Gupta, V: A certain family of summation integral type operators. Math and Comput. Modelling. 37(1213), 1307–1315 (2003). Publisher Full Text

Finta, Z, Gupta, V: Direct and inverse estimates for Phillips type operators. J. Math. Anal. Appl. 303(2), 627–642 (2005). Publisher Full Text

Govil, NK, Gupta, V, Noor, MA: Simultaneous approximation for the Phillips operators. Int. J. Math. Math. Sci. 2006, 1–9 (2006)

Gupta, V: Rate of approximation by new sequence of linear positive operators. Comput. Math. Appl. 45(12), 1895–1904 (2003). Publisher Full Text

Gupta, V, Gupta, MK, Vasishtha, V: Simultaneous approximation by summationintegral type operators. J. Nonlinear Funct. Anal. Appl. 8, 399–412 (2003)

Gupta, V, Maheshwari, P: Bezier variant of a new Durrmeyer type operators. Riv. Mat. Univ. Parma. 7, 9–21 (2003)

May, CP: On Phillips operators. J. Approx. Theory. 20, 315–332 (1977). Publisher Full Text

Ispir, N, Yuksel, I: On the Bezier variant of SrivastavaGupta operators. Applied Math. ENotes. 5, 129–137 (2005)

Deo, N: Faster rate of convergence on SrivastavaGupta operators. Appl. Math. Comput. 218, 10486–10491 (2012). Publisher Full Text

Bai, GD, Hua, YH, Shaw, SY: Rate of approximation for functions with derivatives of bounded variation. Anal. Math. 28(3), 171–196 (2002). Publisher Full Text

Ispir, N, Aral, A, Dogru, O: On Kantorovich process of a sequence of the generalized linear positive operators. Nonlinear Funct. Anal. Optimiz. 29(56), 574–589 (2008). Publisher Full Text

Acar, T, Gupta, V, Aral, A: Rate of convergence for generalized Szász operators. Bull. Math. Sci. 1, 99–113 (2011). Publisher Full Text