On some Statistical Approximation Properties of Generalized Lupas-Stancu Operators

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DOI: 10.46793/KgJMat2205.797Q


The purpose of this paper is to introduce Stancu variant of generalized Lupaş operators whose construction depends on a continuously differentiable, increasing and unbounded function ρ. Depending on the selection of γ and δ, these operators are more flexible than the generalized Lupaş operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation. Finally, we investigate the statistical approximation property of the new operators with the aid of a Korovkin type statistical approximation theorem.


Generalized Lupaş-Stancu operators, Korovkin’s type theorem, convergence theorems, Voronovskaya type theorem, statistical approximation.


[1]   T. Acar, S. Mohiuddine and M. Mursaleen, Approximation by (p,q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory 12 (2018), 1453–1468.

[2]   O. Agratini, On the rate of convergence of a positive approximation process, Nihonkai Math. J. 11 (2000), 47–56.

[3]   S. Bernstein, Démonstation du théorème de Weierstrass fondée sur le calcul de probabilités, Communications of the Kharkov Mathematical Society 13 (1912), 1–2.

[4]   Q.-B. Cai and G. Zhou, On (p,q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput. 276 (2016), 12–20.

[5]   D. Cárdenas-Morales, P. Garrancho and I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (2011), 158–163.

[6]   R. DeVore and G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, Heidelberg, 1993.

[7]   O. Doǧru, G. Içöz and K. Kanat, On the rates of convergence of the q-Lupaş-Stancu operators, Filomat 30 (2016), 1151–1160.

[8]   H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.

[9]   A. Gadjiev, Theorems of the type of P. P. Korovkin’s theorems, Mat. Zametki 20 (1976), 781–786.

[10]   A. Gadjiev and A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comput. Math. Appl. 54 (2007), 127–135.

[11]   A. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. (2002), 129–138.

[12]   A. D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P. P. Korovkin, Dokl. Akad. Nauk 218 (1974), 1001–1004.

[13]   A. Holhos, Quantitative estimates for positive linear operators in weighted spaces, General Mathematics 16 (2008), 99–110.

[14]    H. G. İ. İlarslan, A. Aral and G. Başcanbaz-Tunca, Generalized Lupaş operators, in: AIP Conference Proceedings 1926 1 (2018), Paper ID 020019.

[15]   K. Kanat and M. Sofyalıoğlu, Approximation properties of Stancu-type (p,q)-Baskakov operators, BEU Journal of Science 8 (2019), 889–902.

[16]   K. Kanat and M. Sofyalıoğlu, Some approximation results for Stancu type Lupaş-Schurer operators based on (p,q)-integers, Appl. Math. Comp. 317 (2018), 129–142.

[17]   K. Kanat and M. Sofyalıoğlu, On Stancu type generalization of (p,q)-Baskakov-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68 (2019), 1995–2013.

[18]   A. Karaisa and F. Karakoç, Stancu type generalization of Dunkl analogue of Szàsz operators, Adv. Appl. Clifford Algebr. 26 (2016), 1235–1248.

[19]   P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk 90 (1953), 961–964.

[20]   B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Indagationes Mathematicae (Proceedings) 91 (1988), 53–63.

[21]   G. Lorentz, Bernstein Polynomials, Mathematical Expositions 8, Toronto University Press, Toronto, 1953.

[22]   A. Lupa, The approximation by some positive linear operators, in: Proceedings of the International Dortmund Meeting on Approximation Theory, Witten, Germany, 1995.

[23]   M. Mursaleen and M. Ahasan, The Dunkl generalization of Stancu type q-Szasz-Mirakjan-Kantorovich operators and some approximation results, Carpathian J. Math. 34 (2018), 363–370.

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

[25]   M. Mursaleen, K. J. Ansari and A. Khan, On (p,q)-analogue of Bernstein operators, Appl. Math. Comput. 266(1) (2015), 874–882.

[26]   M. Mursaleen and T. Khan, On approximation by Stancu type jakimovski-Leviatan-Durrmeyer operators, Azerb. J. Math. 7 (2017), 16–26.

[27]   M. Mursaleen, M. Nasiruzzaman, A. Khan and K. J. Ansari, Some approximation results on Bleimann-Butzer-Hahn operators defined by (p,q)-integers, Filomat 30 (2016), 639–648.

[28]   M. Mursaleen, M. Qasim, A. Khan and Z. Abbas, Stancu type q-Bernstein operators with shifted knots, J. Inequal. Appl. 2020 (2020), Article ID 28.

[29]   M. Qasim, M. Mursaleen, A. Khan and Z. Abbas, Approximation by generalized Lupaş operators based on q-integers, Mathematics 8 (2020), Article ID 68.

[30]   S. Rahman, M. Mursaleen and A. Khan, A Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution with error estimation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114(75) (2020), DOI 10.1007/s13398-020-00804-8.

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

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

[33]   K. Weierstrass, Über die analytische darstellbarkeit sogenannter willkürlicher functionen einer reellen veränderlichen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2 (1885), 633–639.