Modified Lupaş-Kantorovich operators with Pólya distribution. (English) Zbl 1528.41034

Agrawal, Gunjan; Gupta, Vijay

Rocky Mt. J. Math. 52, No. 6, 1909-1919 (2022).

Summary: We present a modification of the Kantorovich variant of the operators due to Lupaş and Lupaş which are connected with Pólya distribution and we provide a quantitative Voronovskaya-kind theorem involving modulus of continuity. Also, we observe that a better approximation can be achieved for our modified form of operators. We also estimate the difference of the original operator with its modified form. Lastly, we depict the convergence of the modified operators to some functions using graphical representation.

MSC:

41A25

Rate of convergence, degree of approximation

Keywords:

Pólya distribution; modulus of continuity; asymptotic formula; difference estimate and error estimate

References:

[1]	U. Abel, V. Gupta, and R. N. Mohapatra, “Local approximation by a variant of Bernstein-Durrmeyer operators”, Nonlinear Anal. 68:11 (2008), 3372-3381. · Zbl 1175.41020 · doi:10.1016/j.na.2007.03.026
[2]	R. P. Agarwal and V. Gupta, “On \[q\]-analogue of a complex summation-integral type operators in compact disks”, J. Inequal. Appl. 2012 (2012), art. id. 111. · Zbl 1273.30025 · doi:10.1186/1029-242X-2012-111
[3]	P. N. Agrawal and V. Gupta, “Simultaneous approximation by linear combination of modified Bernstein polynomials”, Bull. Soc. Math. Grèce (N.S.) 30 (1989), 21-29. · Zbl 0747.41014
[4]	P. N. Agrawal, N. Ispir, and A. Kajla, “Approximation properties of Lupaş-Kantorovich operators based on Pólya distribution”, Rend. Circ. Mat. Palermo (2) 65:2 (2016), 185-208. · Zbl 1350.41020 · doi:10.1007/s12215-015-0228-4
[5]	H. Gonska and I. Raşa, “A Voronovskaya estimate with second order modulus of smoothness”, pp. 76-91 in Proc. of the 5th Int. Symposium “Mathematical Inequalities” (Sibiu, Romania), 2008. · Zbl 1212.41016
[6]	H. Gonska, P. Piţul, and I. Raşa, “On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators”, pp. 55-80 in Numerical analysis and approximation theory, Univ. Duisburg, 2006. · Zbl 1123.41012
[7]	V. Gupta, “A note on the rate of convergence of Durrmeyer type operators for function of bounded variation”, Soochow J. Math. 23:1 (1997), 115-118. · Zbl 0869.41016
[8]	V. Gupta, “Differences of operators of Lupaş type”, Constr. Math. Anal. 1:1 (2018), 9-14. · Zbl 1463.41025 · doi:10.33205/cma.452962
[9]	V. Gupta, “On difference of operators with applications to Szász type operators”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113:3 (2019), 2059-2071. · Zbl 1418.30034 · doi:10.1007/s13398-018-0605-x
[10]	V. Gupta and A. M. Acu, “On difference of operators with different basis functions”, Filomat 33:10 (2019), 3023-3034. · Zbl 1499.41030 · doi:10.2298/FIL1910023G
[11]	V. Gupta and O. Duman, “Bernstein-Durrmeyer type operators preserving linear functions”, Mat. Vesnik 62:4 (2010), 259-264. · Zbl 1265.41044
[12]	V. Gupta and N. Ispir, “On simultaneous approximation for some modified Bernstein-type operators”, Int. J. Math. Math. Sci. 71 (2004), 3951-3958. · Zbl 1071.41021 · doi:10.1155/S0161171204312317
[13]	V. Gupta and P. Maheshwari, “Bezier variant of a new Durrmeyer type operators”, Riv. Mat. Univ. Parma (7) 2 (2003), 9-21. · Zbl 1050.41015
[14]	V. Gupta and T. M. Rassias, “Lupaş-Durrmeyer operators based on Polya distribution”, Banach J. Math. Anal. 8:2 (2014), 146-155. · Zbl 1285.41008 · doi:10.15352/bjma/1396640060
[15]	V. Gupta and G. Tachev, “A note on the differences of two positive linear operators”, Constr. Math. Anal. 2:1 (2019), 1-7. · Zbl 1463.41026 · doi:10.33205/cma.469114
[16]	V. Gupta, T. M. Rassias, P. N. Agrawal, and A. M. Acu, Recent advances in constructive approximation theory, Springer Optimization and Its Applications 138, Springer, 2018. · Zbl 1400.41017 · doi:10.1007/978-3-319-92165-5
[17]	V. Gupta, A. M. Acu, and H. M. Srivastava, “Difference of some positive linear approximation operators for higher-order derivatives”, Symmetry 12:6 (2020), art. id. 915. · doi:10.3390/sym12060915
[18]	A. Lupaş, “The approximation by means of some linear positive operators”, pp. 201-229 in Approximation theory (Witten, 1995), Math. Res. 86, Akademie-Verlag, Berlin, 1995. · Zbl 0838.41009
[19]	L. Lupaş and A. Lupaş, “Polynomials of binomial type and approximation operators”, Studia Univ. Babeş-Bolyai Math. 32:4 (1987), 61-69. · Zbl 0659.41018
[20]	D. Miclăuş, “The revision of some results for Bernstein-Stancu type operators”, Carpathian J. Math. 28:2 (2012), 289-300. · Zbl 1289.41038 · doi:10.37193/CJM.2012.02.07
[21]	M. A. Özarslan and O. Duman, “Local approximation behavior of modified SMK operators”, Miskolc Math. Notes 11:1 (2010), 87-99. · Zbl 1224.41056 · doi:10.18514/mmn.2010.228
[22]	D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators”, Rev. Roumaine Math. Pures Appl. 13 (1968), 1173-1194 · Zbl 0167.05001