Approximation by genuine Lupaş-Beta-Stancu operators. (English) Zbl 1388.41016

Summary: In this paper, we introduce a Stancu type generalization of genuine Lupaş-Beta operators of integral type. We establish some moment estimates and the direct results in terms of classical modulus of continuity, Voronovskaja-type asymptotic theorem, weighted approximation, rate of convergence and pointwise estimates using the Lipschitz type maximal function. Lastly, we propose a king type modification of these operators to obtain better estimates.


41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
Full Text: DOI


[1] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer, Berlin (1993). · Zbl 0797.41016
[2] T. Acar, L.N. Mishra and V.N. Mishra, Simultaneous Approximation for Generalized Srivastava-Gupta Operators, Journal of Function Spaces 2015, Article ID 936308, 11 pages. · Zbl 1321.41024
[3] A.D. Gadjiev, Theorems of the type of P. P. korovkin’s theorems, Matematicheskie Zametki 20 (1976), 781-786.
[4] A.D. Gadjiev, R.O. Efendiyev and E. Ibikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J. 1(128) (2003), 45-53. · Zbl 1013.41011
[5] A.R. Gairola, Deepmala and L.N. Mishra, On the q-derivatives of a certain linear positive operators, Iranian Journal of Science and Technology, Transactions A: Science, (2017), DOI 10.1007/s40995-017-0227-8. · Zbl 1397.41005
[6] A.R. Gairola, Deepmala and L.N. Mishra, Rate of Approximation by Finite Iterates of q Durrmeyer Operators, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (April-June 2016) 86:229-234 (2016). doi: 10.1007/s40010-016-0267-z · Zbl 1381.41020
[7] V. Gupta, Th.M. Rassias and E. Pandey, On genuine Lupa¸s-Beta operators and modulus of continuity, Int. J. Nonlinear Anal. Appl. 8 (2017), 23-32. · Zbl 1436.41016
[8] V. Gupta, Th.M. Rassias and R. Yadav, Approximation by Lupa¸s-Beta integral operators, Appl. Math. Comput. 236 (2014), 19-26. · Zbl 1334.41031
[9] V. Gupta and R. Yadav, On approximation of certain integral operators, Acta Math. Viet namica 39 (2014), 193-203. · Zbl 1298.41026
[10] N.K. Govil, V. Gupta and D. Soyba¸s, Certain new classes of Durrmeyer type operators, Appl. Math. Comput. 225 (2013), 195-203. · Zbl 1334.41029
[11] V. Gupta, Th.M. Rassias and J. Sinha, A survey on Durrmeyer operators, Springer In ternational Publishing Switzerland, (2016), 299-312, Contributions in Mathematics and Engineering P.M. Pardalos, T.M. Rassias (eds.). · Zbl 1371.41034
[12] G.C. Jain, Approximation of functions by a new class of linear operators, J. Aust. Math. Soc. 13 (1972), 271-276. · Zbl 0232.41003
[13] A. Kumar, Voronovskaja type asymptotic approximation by general Gamma type opera tors, Int. J. of Mathematics and its Applications 3 (2015), 71-78.
[14] A. Kumar,Approximation by Stancu type generalized Srivastava-Gupta operators based on certain parameter, Khayyam J. Math. 3, no. 2 (2017), pp. 147-159. DOI: 10.22034/kjm.2017.49477 · Zbl 1384.41018
[15] A. Kumar, General Gamma type operators in L p spaces, Palestine Journal of Mathematics 7 (2018), 73-79. · Zbl 1375.41012
[16] A. Kumar and D. K. Vishwakarma, Global approximation theorems for general Gamma type operators, Int. J. of Adv. in Appl. Math. and Mech. 3(2) (2015), 77-83. · Zbl 1359.41007
[17] A. Kumar and L.N. Mishra, Approximation by modified Jain-Baskakov-Stancu operators, Tbilisi Mathematical Journal 10 (2017), pp. 185-199. · Zbl 1371.41021
[18] A. Kumar, V.N. Mishra and Dipti Tapiawala, Stancu type generalization of modified Srivastava-Gupta operators, Eur. J. Pure Appl. Math 10, (2017), 890-907. · Zbl 1370.41027
[19] A. Kumar, Artee and D.K. Vishwakarma, Approximation properties of general gamma type operators in polynomial weighted space, Int. J. Adv. Appl. Math. and Mech. 4 (2017), pp. 7-13. · Zbl 1390.41032
[20] J.P. King, Positive linear operators which preserve x 2, Acta Math. Hungar. 99 (2003), 203-208. · Zbl 1027.41028
[21] B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl. Akad. Indag. Math. 50 (1988), 53-63. · Zbl 0652.42004
[22] N.I. Mahmudov and V. Gupta, On certain q-analogue of S´zasz Kantorovich operators, J. Appl. Math. Comput. 37 (2011), 407-419. · Zbl 1294.41015
[23] V.N. Mishra, P. Sharma and M. Birou, Approximation by Modified Jain-Baskakov Oper ators, arXiv:1508.05309v2 [math.FA] 9 Sep 2015. 28 Alok Kumar, Vandana
[24] V.N. Mishra and P. Sharma, On approximation properties of Baskakov-Schurer-S´zasz operators, arXiv:1508.05292v1 [math.FA] 21 Aug 2015.
[25] V.N. Mishra, H.H. Khan, K. Khatri and L.N. Mishra, Hypergeometric Representation for Baskakov-Durrmeyer-Stancu Type Operators, Bulletin of Mathematical Analysis and Applications 5 Issue 3 (2013), Pages 18-26. · Zbl 1314.41013
[26] V.N. Mishra, K. Khatri and L.N. Mishra, On Simultaneous Approximation for Baskakov Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Physical Sciences 24, 3(A) (2012), pp. 567-577. · Zbl 1339.41021
[27] V.N. Mishra, K. Khatri, L.N. Mishra and Deepmala, Inverse result in simultaneous approx imation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications (2013), 2013:586. doi:10.1186/1029-242X-2013-586. · Zbl 1295.41013
[28] V.N. Mishra, K. Khatri and L.N. Mishra, Some approximation properties of q-Baskakov Beta-Stancu type operators, Journal of Calculus of Variations 2013, Article ID 814824, 8 pages. · Zbl 1298.41039
[29] V.N. Mishra, Rajiv B. Gandhi and Ram N. Mohapatraa, Summation-Integral type modifi cation of S´zasz-Mirakjan-Stancu operators, J. Numer. Anal. Approx. Theory 45(1) (2016), pp. 27-36. · Zbl 1399.41047
[30] V.N. Mishra, K. Khatri, L.N. Mishra and Deepmala, Inverse result in simultaneous ap proximation by Baskakov-Durrmeyer-Stancu operators, J. Inequal. Appl. 586 (2013), 1-11. · Zbl 1295.41013
[31] M.A. ¨Ozarslan and H. Aktu˘glu, Local approximation for certain King type operators, Filomat 27 (2013), 173-181. · Zbl 1458.41008
[32] E. Pandey and R.K. Mishra, Convergence estimates in simultaneous approximation by certain Srivastava-Gupta type operators, Adv. Studies Contemporary Math. 26 (2016), 467-480. · Zbl 1354.41022
[33] D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl. 13 (1968), 1173-1194. · Zbl 0167.05001
[34] O. Sz´asz, Generalization of S. Bernstein polynomials to the infinite interval, J. Res. Natl. Bur. Stand. 45 (1950), 239-245.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.