Geodesic
Approximation by modified Lupa\c{s}-Stancu operators based on $(p, q)$-integers
Eurasian mathematical journal, Tome 12 (2021) no. 2, pp. 39-51.

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The purpose of this paper is to construct a new class of Lupaş operators in the frame of post quantum setting. We obtain a Korovkin type approximation theorem, study the rate of convergence of these operators by using the concept of the $K$-functional and modulus of continuity, also give a convergence theorem for the Lipschitz continuous functions. 
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A. Khan; Z. Abbas; M. Qasim; M. Mursaleen. Approximation by modified Lupa\c{s}-Stancu operators based on $(p, q)$-integers. Eurasian mathematical journal, Tome 12 (2021) no. 2, pp. 39-51. http://geodesic.mathdoc.fr/item/EMJ_2021_12_2_a4/

[1] T. Acar, A. Aral, S. A. Mohiuddine, “Approximation by bivariate (p, q)-Bernstein-Kantorovich operators”, Iran. J. Sci. Technol. Trans., Sect. A: Science, 42 (2018), 655–662 | DOI | MR | Zbl

[2] T. Acar, S. A. Mohiudine, M. Mursaleen, “Approximation by (p, q)-Baskakov-Durrmeyer-Stancu operators”, Complex Anal. Oper. Theory, 12:6 (2018), 1453–1468 | DOI | MR | Zbl

[3] H. Ben Jebreen, M. Mursaleen, M. Ahasan, “On the convergence of Lupaş (p, q)-Bernstein operators via contraction principle”, J. Inequal. Appl., 34 (2019) | DOI | MR

[4] S. N. Bernstein, “Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités”, Commun. Soc. Math. Kharkow, 13:1–2 (1912–1913)

[5] A. R. Devdhara, V. N. Mishra, “Stancu variant of (p, q)-Szász-Mirakyan operators”, J. Inequal. Special Functions, 8:5 (2017), 1–7 | MR

[6] R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1993 | MR | Zbl

[7] A. D. Gadjiev, R. O. Efendiyev, E. Ibikli, “On Korovkin type theorem in the space of locally integrable functions”, Czechoslovak Math. J., 53:1 (2003), 45–53 | DOI | MR | Zbl

[8] A. R. Gairola, Deepmala, L. N. Mishra, “On the q-derivatives of a certain linear positive operators”, Iran. J. Sci. Technol. Trans., Sect. A: Science, 42:3 (2018), 1409–1417 | DOI | MR | Zbl

[9] A. R. Gairola, Deepmala, L. N. Mishra, “Rate of approximation by finite iterates of q-Durrmeyer operators”, Proc. Natl. Acad. Sci., India, Sect. A: Phys. Sci, 86:2 (2016), 229–234 | DOI | MR | Zbl

[10] E. Ibikli, E. A. Gadjieva, “The order of approximation of some unbounded functions by the sequence of positive linear operators”, Turkish J. Math., 19:3 (1995), 331–337 | MR | Zbl

[11] U. Kadak, V. N. Mishra, S. Pandey, “Chlodowsky type generalization of (p, q)-Szász operators involving Brenke type polynomials”, Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Series A. Matematicas (RACSAM), 112:4 (2018), 1443–1462 | DOI | MR | Zbl

[12] K. Khan, D. K. Lobiyal, “Bézier curves based on Lupaş (p, q)-analogue of Bernstein functions in CAGD”, J. Comput. Appl. Math., 317 (2017), 458–477 | DOI | MR | Zbl

[13] K. Khan, D. K. Lobiyal, A. Kilicman, “A de Casteljau algorithm for Bernstein type polynomials based on (p, q)-integers”, Appl. Appl. Math., 13:2 (2018), 997–1017 | MR | Zbl

[14] K. Khan, D. K. Lobiyal, A. Kilicman, “Bézier curves and surfaces based on modified Bernstein polynomials”, Azerb. J. Math., 9:1 (2019), 1–19 | MR

[15] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Co., Delhi, 1960 | MR

[16] B. Lenze, “On Lipschitz type maximal functions and their smoothness spaces”, Nederl. Akad. Indag. Math., 50 (1988), 53–63 | DOI | MR | Zbl

[17] A. Lupaş, “The approximation by some positive linear operators”, Proceedings of the International Dortmund meeting on Approximation Theory, eds. M. W. Muller et al., Akademie Verlag et al., Berlin, 1995, 201–229 | MR | Zbl

[18] L. N. Mishra, S. Pandey, V. N. Mishra, “On a class of generalised (p, q) Bernstein operators”, Indian J. Industrial App. Mathematics, 10:12 (2019), 220–233 | DOI

[19] V. N. Mishra, A. R. Devdhara, “On Stancu type generalization of (p, q)-Szász-Mirakyan Kantorovich type operators”, J. Appl. Math. Inf., 36:3 (2018), 285–299 | MR | Zbl

[20] V. N. Mishra, K. Khatri, L. N. Mishra, Deepmala, “Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators”, J. Inequal. Appl., 586 (2013) | DOI | MR | Zbl

[21] V. N. Mishra, M. Mursaleen, S. Pandey, A. Alotaibi, “Approximation properties of Chlodowsky variant of (p, q) Bernstein-Stancu-Schurer operators”, J. Inequal. Appl., 176 (2017) | DOI | MR | Zbl

[22] V. N. Mishra, S. Pandey, “On (p, q) Baskakov-Durrmeyer-Stancu operators”, Adv. Appl. Clifford Algebra, 27:2 (2017), 1633–1646 | DOI | MR | Zbl

[23] V. N. Mishra, S. Pandey, “On Chlodowsky variant of (p, q) Kantorovich-Stancu-Schurer operators”, Int. J. Anal. Appl., 11:1 (2016), 28–39 | Zbl

[24] M. Mursaleen, M. Ahasan, K. J. Ansari, “Bivariate Bernstein-Schurer-Stancu type GBS operators in (p, q)-analogue”, Adv. Difference Equ., 76 (2020), 1–17 | MR

[25] M. Mursaleen, K. J. Ansari, A. Khan, “On (p, q)-analogue of Bernstein operators”, Appl. Math. Comput., 266 (2015), 874–882 ; Erratum: Appl. Math. Comput., 278 (2016), 70–71 | MR | Zbl | MR | Zbl

[26] M. Mursaleen, K. J. Ansari, A. Khan, “Some approximation results by (p,q)-analogue of Bernstein-Stancu operators”, Appl. Math. Comput., 264 (2015), 392–402 | MR | Zbl

[27] M. Mursaleen, M. Qasim, A. Khan, Z. Abbas, “Stancu type q-Bernstein operators with shifted knots”, J. Inequl. Appl., 1 (2020), 1–14 | MR

[28] M. Mursaleen, Md. Nasiruzzaman, A. Khan, K. J. Ansari, “Some approximation results on Bleimann-Butzer-Hahn operators defined by (p, q)-integers”, Filomat, 30:3 (2016), 639–648 | DOI | MR | Zbl

[29] M. Qasim, M. Mursaleen, A. Khan, Z. Abbas, “Approximation by generalized Lupaş operators based on q-integers”, Mathematics, 8:1 (2020), 1–15 | DOI

[30] N. Rao, A. Wafi, “(p, q)-Bivariate-Bernstein-Chlowdosky operators”, Filomat, 32:2 (2018), 369–378 | DOI | MR

[31] N. Rao, A. Wafi, “Bivariate-Schurer-Stancu operators based on (p, q)-integers”, Filomat, 32:4 (2018), 1251–1258 | DOI | MR

[32] D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators”, Rev. Roumaine Math. Pure Appl., 13 (1968), 1173–1194 | MR | Zbl

[33] K. K. Sing, A. R. Gairola, Deepmala, “Approximation theorems for q-analogue of a linear operator by A. Lupaş”, Int. Jour. Anal. Appl., 1:12 (2016), 30–37 | Zbl

[34] K. Weierstrass, “Uber die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen”, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885, 633–639