Convergence in simultaneous approximation for Srivastava-Gupta operators

  • Durvesh Kumar Verma1Email author and

    Affiliated with

    • Purshottam N Agrawal1

      Affiliated with

      Mathematical Sciences20126:22

      DOI: 10.1186/2251-7456-6-22

      Received: 18 May 2012

      Accepted: 12 July 2012

      Published: 16 August 2012

      Abstract

      Purpose

      The purpose of the present paper is to introduce the generalized form of Srivastava-Gupta 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 Srivastava-Gupta operators Simultaneous approximation Rate of convergence 2010 26A45 41A28

      Introduction

      In the year 2003, Srivastava and Gupta [1] 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 [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 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 G n,c is defined as follows:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ1_HTML.gif
      (1.1)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ2_HTML.gif
      (1.2)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equa_HTML.gif
      Here http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq1_HTML.gif is a sequence of functions defined on the closed interval [0,b, b>0, satisfying the following properties. For each http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq2_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq3_HTML.gif :
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq4_HTML.gif ,

         
      2. (ii)

        ϕ n,c (0)=1,

         
      3. (iii)

        ϕ n,c is completely monotone so that http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq5_HTML.gif and

         
      4. (iv)
        there exists an integer c such that
        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equb_HTML.gif
         

      (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 Baskakov-Durrmeyer 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 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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq6_HTML.gif 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
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ3_HTML.gif
      (1.3)

      where p n,k (x c) 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:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equc_HTML.gif

      where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq7_HTML.gif .

      We denote that the class of absolutely continuous functions f on (0,) by D B q (0,), (where q is some positive integer) are satisfied:
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq8_HTML.gif and

         
      2. (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 fD B q (0,), we can have the representation

         
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equd_HTML.gif

      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 Srivastava-Gupta operators.

      Results and discussion

      In the sequel we shall need the following lemmas:

      Lemma 1

      If we define the moments as
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Eque_HTML.gif
      and then, T r,n,0(x,c)=1, http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq9_HTML.gif 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)
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equf_HTML.gif
      Consequently,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equg_HTML.gif
      Furthermore, T n,r,m (x,c) is polynomial of degree m in x and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equh_HTML.gif

      Proof

      Taking the derivative of T n,r,m (x,c) with respect to x and using the identity http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq10_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equi_HTML.gif
      To compute I 2 we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equj_HTML.gif
      Using http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq11_HTML.gif p n−(r−1)c,k (t,c), we can write I 1 as
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equk_HTML.gif
      Again using t(tx) m =(tx) m + 1 + x(tx) m and integrating by parts, we get
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equl_HTML.gif
      Proceeding in a similar manner, we obtain J 2 as
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equm_HTML.gif
      Combining I 1,I 2,J 1, and J 2, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equn_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equo_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equp_HTML.gif

      Remark 2

      Let x∈(0,) and λ>2; then for n sufficiently large, Lemma 1 yields that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equq_HTML.gif

      Lemma 2

      Let x∈(0,) and λ>2; then for n sufficiently large, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equr_HTML.gif

      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
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equs_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq12_HTML.gif and n>max{q,r + s + 1}, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equt_HTML.gif

      Proof

      We prove this result by applying the principle of mathematical induction and using the following identity:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ4_HTML.gif
      (2.1)
      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
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equu_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equv_HTML.gif
      which shows that the result holds for s=1. Let us suppose that the result holds for s=m i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equw_HTML.gif

      Now by Equation 2.1,

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq13_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equx_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equy_HTML.gif
      Integrating by parts the last integral, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equz_HTML.gif
      Therefore,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equaa_HTML.gif

      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 fD B q (0,), q>0 and x∈(0,). The for λ>2 and n sufficiently large, we have http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq14_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equab_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq15_HTML.gif denotes the total variation of f x on [a,b], and the auxiliary function f x is defined by
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equac_HTML.gif

      Proof

      Using the mean value theorem, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equad_HTML.gif
      Also it is a valid identity that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equae_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equaf_HTML.gif
      Obviously, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equag_HTML.gif

      Thus, using the above identities, we can write

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq16_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ5_HTML.gif
      (3.1)
      Also it can be verified that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ6_HTML.gif
      (3.2)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ7_HTML.gif
      (3.3)

      Combining Equations 3.1–3.3, we get

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq17_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equah_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ8_HTML.gif
      (3.4)

      Applying Remark 2 and Lemma 1 in Equation 3.4, we have

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq18_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ9_HTML.gif
      (3.5)
      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:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equai_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equaj_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ10_HTML.gif
      (3.6)
      For estimating the integral http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq19_HTML.gif above, we proceed as follows: since t≥2x implies that t≤2(tx) Schwarz inequality which follows from Lemma 1,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ11_HTML.gif
      (3.7)
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ12_HTML.gif
      (3.8)
      By using the Hölder’s inequality and Remark 2, we get the estimate as follows:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equak_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ13_HTML.gif
      (3.9)
      Collecting the estimates from Equations 3.6–3.9, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ14_HTML.gif
      (3.10)
      On the other hand, to estimate B n,r (f,x) by applying the Lemma 2 with http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq20_HTML.gif and integration by parts, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equal_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equ15_HTML.gif
      (3.11)

      where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq21_HTML.gif .

      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 D B q (0,), q>0 and x∈(0,). The for λ>2 and n sufficiently large, we have

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq22_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equam_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_IEq23_HTML.gif denotes the total variation of f x on [a,b], and the auxiliary function D s + 1 f x is defined by
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-22/MediaObjects/40096_2012_24_Equan_HTML.gif

      Conclusions

      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.

      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.

      Declarations

      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.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Indian Institute of Technology Roorkee

      References

      1. Srivastava HM, Gupta V: A certain family of summation integral type operators. Math and Comput. Modelling 2003,37(12–13):1307–1315.MathSciNetMATHView Article
      2. Finta Z, Gupta V: Direct and inverse estimates for Phillips type operators. J. Math. Anal. Appl 2005,303(2):627–642.MathSciNetMATHView Article
      3. Govil NK, Gupta V, Noor MA: Simultaneous approximation for the Phillips operators. Int. J. Math. Math. Sci 2006, 2006:1–9.MathSciNetView Article
      4. Gupta V: Rate of approximation by new sequence of linear positive operators. Comput. Math. Appl 2003,45(12):1895–1904.MathSciNetMATHView Article
      5. Gupta V, Gupta MK, Vasishtha V: Simultaneous approximation by summation-integral type operators. J. Nonlinear Funct. Anal. Appl 2003, 8:399–412.MathSciNetMATH
      6. Gupta V, Maheshwari P: Bezier variant of a new Durrmeyer type operators. Riv. Mat. Univ. Parma 2003, 7:9–21.MathSciNet
      7. May CP: On Phillips operators. J. Approx. Theory 1977, 20:315–332.MATHView Article
      8. Ispir N, Yuksel I: On the Bezier variant of Srivastava-Gupta operators. Applied Math. E-Notes 2005, 5:129–137.MathSciNetMATH
      9. Deo N: Faster rate of convergence on Srivastava-Gupta operators. Appl. Math. Comput 2012, 218:10486–10491.MathSciNetMATHView Article
      10. Bai GD, Hua YH, Shaw SY: Rate of approximation for functions with derivatives of bounded variation. Anal. Math 2002,28(3):171–196.MathSciNetMATHView Article
      11. Ispir N, Aral A, Dogru O: On Kantorovich process of a sequence of the generalized linear positive operators. Nonlinear Funct. Anal. Optimiz 2008,29(5–6):574–589.MathSciNetMATHView Article
      12. Acar T, Gupta V, Aral A: Rate of convergence for generalized Szász operators. Bull. Math. Sci 2011, 1:99–113.MathSciNetMATHView Article

      Copyright

      © Verma and Agrawal; licensee Springer. 2012

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.