Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators
© Mishra et al.; licensee Springer. 2013
Received: 15 July 2013
Accepted: 29 October 2013
Published: 17 December 2013
In the present paper, we study an inverse result in simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators.
MSC:41A25, 41A35, 41A36.
where and .
For , these operators reduce to Baskakov-Durrmeyer operators . Note that this case was investigated in . Several other researchers have studied in this direction and obtained different approximation properties of many operators, and we mention some of them as [3–8]etc. Verma et al.  also studied some approximation properties, asymptotic formula and better estimates for these operators. Recently, Gupta et al.  and Mishra and Khatri  established point-wise convergence, a Voronovskaja-type asymptotic formula and an error estimate in terms of modulus of continuity of the function and investigated moments of these operators using hypergeometric series, errors estimation in simultaneous approximation, respectively.
Let , where , be the class of all continuous functions defined on satisfying the growth condition . The norm on is defined as .
where is k th forward difference with step length h.
2 Auxiliary results
In the sequel we shall need several lemmas.
Lemma 1 
Lemma 2 
Lemma 3 
Consequently, , where is an integral part of m.
Lemma 4 
Lemma 5 
Consequently, , i.e., , where and are some positive constants.
Combining the estimates (2.5)-(2.9), we get (2.4). The other consequence follows from . This completes the proof of the lemma. □
Lemma 7 
Let and with . If , then .
3 Known and inverse results
In this section, first we give some known results and then we estimate an inverse theorem in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. Now, this section is devoted to the following inverse theorem in simultaneous approximation.
Theorem 1 
Theorem 2 
Theorem 3 
where , .
where denotes the Zygmund class satisfying .
Thus by Lemmas 5 and 7, we have also on the interval , and it proves that . This completes the validity of the implication for the case .
Since , therefore by Lemmas 6 and 7, we have also on the interval , which proves that . This completes the validity of the implication for the case . This completes the proof of the theorem. □
Dr. VNM is an assistant professor at the Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat, Gujarat, India and he is a very active researcher in various fields of mathematics like Approximation theory, summability theory, variational inequalities, fixed point theory and applications, operator analysis, nonlinear analysis etc. A Ph.D. in Mathematics, he is a double gold medalist, ranking first in the order of merit in both B.Sc. and M.Sc. Examinations from the Dr. Ram Manohar Lohia Avadh University, Faizabad (Uttar Pradesh), India. Dr. VNM has undergone rigorous training from IIT, Roorkee, Mumbai, Kanpur, ISI Banglore in computer oriented mathematical methods and has experience of teaching post graduate, graduate and engineering students. Dr. VNM has to his credit many research publications in reputed journals including SCI/SCI(Exp.) accredited journals. Dr. VNM is referee of several international journals in the frame of pure and applied mathematics and Editor of reputed journals covering the subject mathematics. The second author KK is a research scholar (R/S) in Applied Mathematics and Humanities Department at the Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat (Gujarat), India under the guidance of Dr. VNM and expert in operator analysis. Recently the third author LNM joined as a full-time research scholar (FIR) at the Department of Mathematics, National Institute of Technology, Silchar-788010, District-Cachar, Assam, India and he is also very good active researcher in approximation theory, summability analysis, integral equations, nonlinear analysis, optimization technique, fixed point theory & applications and operator theory. The fourth author Deepmala is referee of many journals like as Elsevier Journals, Bull. Math. Anal. Appl. Demonstratio Mathematica, African J. Math. Math. Sci. etc. and she is very expert in Integral Equations, Non-linear analysis, dynamic programming, Fixed point theory and applications etc.
The authors wish to express their gratitude to the anonymous referees for their detailed criticism and elaborate suggestions which have helped them to improve the paper substantially. They have thus been able to eliminate some mistakes and to present the manuscript in a more compact manner. Authors mention their sense of gratitude to their great master and friend academician Prof. Ravi P. Agarwal, Texas A and M University-Kingsville, TX, USA, for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely as well as for supporting this work. The authors are also thankful to all the editorial board members and reviewers of prestigious Science Citation Index (SCI) journal, i.e., Journal of Inequalities and Applications (JIA). This research article is totally supported by CPDA, SVNIT, Surat (Gujarat), India.
- Verma DK, Gupta V, Agrawal PN: Some approximation properties of Baskakov-Durrmeyer-Stancu operators. Appl. Math. Comput. 2012, 218: 6549–6556. 10.1016/j.amc.2011.12.031MATHMathSciNetView ArticleGoogle Scholar
- Finta Z: On converse approximation theorems. J. Math. Anal. Appl. 2005, 312: 159–180. 10.1016/j.jmaa.2005.03.044MATHMathSciNetView ArticleGoogle Scholar
- Agrawal PN, Thamer KJ: Approximation of unbounded functions by a new sequence of linear positive operators. J. Math. Anal. Appl. 1998, 225: 660–672. 10.1006/jmaa.1998.6036MATHMathSciNetView ArticleGoogle Scholar
- Gupta V, Noor MA, Beniwal MS, Gupta MK: On simultaneous approximation for certain Baskakov-Durrmeyer type operators. J. Inequal. Pure Appl. Math. 2006., 7: Article ID 125Google Scholar
- Gupta MK, Gupta V, Kumar M: Direct and inverse estimates for a new family of linear positive operators. J. Math. Anal. Appl. 2007, 330: 799–816. 10.1016/j.jmaa.2006.08.012MATHMathSciNetView ArticleGoogle Scholar
- Govil NK, Gupta V: Direct estimates in simultaneous approximation for Durrmeyer type operators. Math. Inequal. Appl. 2007, 10: 371–379.MATHMathSciNetGoogle Scholar
- Gupta V: Some approximation properties on q -Durrmeyer operators. Appl. Math. Comput. 2008, 197: 172–178. 10.1016/j.amc.2007.07.056MATHMathSciNetView ArticleGoogle Scholar
- Gupta V, Yadav R: Direct estimates in simultaneous approximation for BBS operators. Appl. Math. Comput. 2012, 218: 11290–11296. 10.1016/j.amc.2012.05.021MATHMathSciNetView ArticleGoogle Scholar
- Gupta V, Verma DK, Agrawal PN: Simultaneous approximation by certain Baskakov-Durrmeyer-Stancu operators. J. Egypt. Math. Soc. 2012, 20(3):183–187. 10.1016/j.joems.2012.07.001MATHMathSciNetView ArticleGoogle Scholar
- Mishra VN, Khan HH, Khatri K, Mishra LN: Hypergeometric representation for Baskakov-Durrmeyer-Stancu type operators. Bull. Math. Anal. Appl. 2013, 5(3):18–26.MathSciNetGoogle Scholar
- Gupta V: A note on modified Baskakov type operators. Approx. Theory Its Appl. 1994, 10: 74–78.MATHGoogle Scholar
- Berens H, Lorentz GG: Inverse theorem for Bernstein polynomials. Indiana Univ. Math. J. 1972, 21: 693–708. 10.1512/iumj.1972.21.21054MATHMathSciNetView ArticleGoogle Scholar
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.