Abstract
In the present paper, we study an inverse result in simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators.
MSC:41A25, 41A35, 41A36.
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1 Introduction
Verma et al. [1] considered Baskakov-Durrmeyer-Stancu (abbr. BDS) operators for as
where and .
For , these operators reduce to Baskakov-Durrmeyer operators . Note that this case was investigated in [2]. 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. [1] also studied some approximation properties, asymptotic formula and better estimates for these operators. Recently, Gupta et al. [9] and Mishra and Khatri [10] 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 .
Let
here being a type of the Dirac delta function. Then operators (1.1) can be written in the following form:
The operators are well defined for . It is easily checked that the operators defined above are linear positive operators and . It turns out that the order of approximation by these operators is at best as , howsoever smooth the function f cab is. Throughout this paper, we denote by the space of all continuous functions on the interval , the norm denotes the sup norm on the space . For and a positive integer , the k th order modulus of continuity is defined as
where is k th forward difference with step length h.
A function f is said to belong to the generalized Zygmund class if for there exists a constant C such that . In particular for , we simply write instead of . By we mean the class of continuous functions defined on having a compact support and the subclass of , consisting of s-times continuously differentiable functions with and . Also let
For with , Peetre’s K-functionals are defined as
For and with , we say that if
2 Auxiliary results
In the sequel we shall need several lemmas.
Lemma 1 [10]
For , and , we have
Moreover,
Lemma 2 [10]
For and , we have
Lemma 3 [11]
For , if
then , , and we have the recurrence relation:
Consequently, , where is an integral part of m.
Lemma 4 [1]
For , if
then
and for we have the recurrence relation:
From the recurrence relation, it is easily verified that for all , we have
Lemma 5 [11]
There exist polynomials on , independent of n and k, such that
Lemma 6 Let and with . If
then
Consequently, , i.e., , where and are some positive constants.
Proof To prove (2.3), it is sufficient to show that
Since , therefore by Theorem 2 there exists a function such that for and ,
which implies that
Thus, it is sufficient to show that there exits a constant such that for each ,
In fact, by the linearity property, we have
Applying Lemma 5, we have
Therefore, by the Cauchy-Schwarz inequality and Lemma 3, we get
where the constant is independent of f and g. Next, by Taylor’s expansion, we have
where ξ lies between t and x. Using the above expansion and the fact that
we get
Also, by Lemmas 3, 4 and 5 and the Cauchy-Schwarz inequality, we have
Hence
Combining the estimates (2.5)-(2.9), we get (2.4). The other consequence follows from [12]. This completes the proof of the lemma. □
Lemma 7 [5]
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 [9]
If , for some , and exists at a point , then
Theorem 2 [9]
Let for some , and exists at a point . Then
Theorem 3 [9]
Let for some , and . Then, for sufficiently large n, we have
where , .
Theorem 4 Let , , and suppose . Then in the following statements :
-
(i)
,
-
(ii)
,
where denotes the Zygmund class satisfying .
Proof Let us choose , , , in such a way that . Also suppose with and on the interval . For with , we have
By the Leibniz formula, we have
Applying Theorem 3, we have
uniformly in . By Taylor’s expansion of and , we have
and
Substituting the above expansions in and using Theorem 2, the Schwarz inequality and Lemma 4, we obtain
uniformly in . Again using the Leibniz formula, we have
uniformly in . Combining the above estimates, we get
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 .
To prove the result for for any interval , let , be such that and . Letting we shall prove the assertion . From the previous case it implies that exists and belongs to . Let be such that on the interval and . If denotes the characteristic function of the interval , we have
Using the linearity property, the Leibniz formula and Theorem 3, we have
Applying the Leibniz formula and Theorem 2, we get
Then by Theorem 3, we have
uniformly in . Applying Taylor’s expansion of , we have
where ξ lies between t and x. Using Theorem 2, we get
Again using Taylor’s expansion of and using the fact that as , we have
Since for every , therefore by Theorem 2 and Lemma 2, we have
uniformly in . Also as in the proof of Theorem 1, it can be easily shown that
uniformly in . Next, using Lemma 5, the mean value theorem, the Schwarz inequality and Lemma 4, we have
where η lies between t and x, and choose δ such that . Combining the above estimates, we get
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. □
Authors’ information
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.
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Acknowledgements
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.
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Authors’ contributions
VNM, KK, LNM and Deepmala computed the auxiliary results and inverse theorem in simultaneous approximation for Baskakov-Durrmeyer-Stancu Operators. VNM and Deepmala conceived of the study and participated in its design and coordination. VNM, KK, LNM and Deepmala contributed equally and significantly in writing this manuscript. All the authors drafted the manuscript, read and approved the final version of manuscript in JIA.
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Mishra, V.N., Khatri, K., Mishra, L.N. et al. Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators. J Inequal Appl 2013, 586 (2013). https://doi.org/10.1186/1029-242X-2013-586
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DOI: https://doi.org/10.1186/1029-242X-2013-586