Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

References

• Weierstrass, V. K. (1885). Ueber die analytische Darstellbarkeit sogennanter willkürlicher Functionen einer reellen Veranderlichen. In: Sitzungsberichte Der Akademie zu Berlin. pp. 633–639.
   Google Scholar
• Bernstein, S. N. (1912). Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. Soc. Math. Kharkow. 13:1–2.
   Google Scholar
• Bernstein, S. N. (1912). Sur les recherches récentes relatives à la meilleure approximation des fonctions continues par les polynomes. Proc. of 5th Inter. Math. Congress. 1:256–266.
   Google Scholar
• Agrawal, P. N., Finta, Z., Kumar, A. S. (2015). Bernstein-Schurer-Kantorovich operators based on q-integers. Appl. Math. Comput. 256:222–231. DOI: 10.1016/j.amc.2014.12.106.
   Web of Science ®Google Scholar
• Cai, Q. B. (2018). The Bézier variant of Kantorovich type λ-Bernstein operators. J. Inequal. Appl. 2018(1):90. DOI: 10.1186/s13660-018-1688-9.
   PubMed Web of Science ®Google Scholar
• Cal, J., Valle, A. M. (2000). A generalization of Bernstein-Kantorovich operators. J. Math. Anal. Appl. 252(2):750–766. DOI: 10.1006/jmaa.2000.7119.
   Web of Science ®Google Scholar
• Il'inskii, A., Ostrovska, S. (2002). Convergence of generalized Bernstein polynomials. J. Approx. Theory. 116 (1):100–112. DOI: 10.1006/jath.2001.3657.
   Web of Science ®Google Scholar
• Kadak, U. (2017). On relative weighted summability in modular function spaces and associated approximation theorems. Positivity. 21(4):1593–1614. DOI: 10.1007/s11117-017-0487-8.
   Web of Science ®Google Scholar
• Kadak, U., Mishra, V. N., Pandey, S. (2018). Chlodowsky type generalization of (p,q)-Szasz operators involving Brenke type polynomials. RACSAM. 112(4):1443–1462. DOI: 10.1007/s13398-017-0439-y.
   Web of Science ®Google Scholar
• Kucukoglu, I., Simsek, B., Simsek, Y. (2019). Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature. Appl. Math. Comput. 344–345:150–162. DOI: 10.1016/j.amc.2018.10.012.
   Web of Science ®Google Scholar
• Wu, Z., Zhou, X. (2020). Polynomial convergence order of stochastic Bernstein approximation. Adv. Comput. Math. 46:8.
   Web of Science ®Google Scholar
• Chen, X., Tan, J., Liu, Z., Xie, J. (2017). Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450(1):244–261. DOI: 10.1016/j.jmaa.2016.12.075.
   Web of Science ®Google Scholar
• Cai, Q. B., Lian, B.-Y., Zhou, G. (2018). Approximation properties of λ-Bernstein operators. J. Inequal. Appl. 2018(1):61. DOI: 10.1186/s13660-018-1653-7.
   PubMed Web of Science ®Google Scholar
• Acu, A.-M., Manav, N., Sofonea, D. F. (2018). Approximation properties of λ-Kantorovich operators. J. Inequal. Appl. 2018(1):202. DOI: 10.1186/s13660-018-1795-7.
   PubMed Web of Science ®Google Scholar
• Khosravian-Arab, H., Dehghan, M., Eslahchi, M. R. (2018). A new approach to improve the order of approximation of the Bernstein operators: Theory and applications. Numer. Algor. 77(1):111–150. DOI: 10.1007/s11075-017-0307-z.
   Web of Science ®Google Scholar
• Mohiuddine, S. A., Özger, F. (2020). Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α. RACSAM. 114(2):70. DOI: 10.1007/s13398-020-00802-w.
   Web of Science ®Google Scholar
• Mohiuddine, S. A., Acar, T., Alotaibi, A. (2017). Construction of a new family of Bernstein-Kantorovich operators. Math. Meth. Appl. Sci. 40(18):7749–7759. DOI: 10.1002/mma.4559.
   Web of Science ®Google Scholar
• Kantorovich, L. (1930). Sur Certain Developpements Suivant les Polynomes de La Forme de S. Bernstein, I, II. C. R. Acad. URSS. 563–568:595–600.
   Google Scholar
• Stancu, D. D. (1983). Approximation of functions by means of a new generalized Bernstein operator. Calcolo. 20(2):211–229. DOI: 10.1007/BF02575593.
   Google Scholar
• Acu, A.-M. (2015). Stancu-Schurer-Kantorovich operators based on q-integers. Appl. Math. Comput. 259:896–907. DOI: 10.1016/j.amc.2015.03.032.
   Web of Science ®Google Scholar
• Chauhan, R., İspir, N., Agrawal, P. (2017). A new kind of Bernstein-Schurer-Stancu-Kantorovich-type operators based on q-integers. J. Inequal. Appl. 2017(1):50. DOI: 10.1186/s13660-017-1298-y.
   PubMed Web of Science ®Google Scholar
• Gadjiev, A. D., Ghorbanalizadeh, A. M. (2010). Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables. Appl. Math. Comput. 216(3):890–901.
   Web of Science ®Google Scholar
• Kanat, K., Sofyalıoğlu, M. (2018). Some approximation results for Stancu type Lupas-Schurer operators based on (p,q)-integersAppl. Math. Comput. 317:129–142. DOI: 10.1016/j.amc.2017.08.046.
   Web of Science ®Google Scholar
• Oz, G. (2012). A Kantorovich variant of a new type Bernstein-Stancu polynomials. Appl. Math. Comput. 218 (17):8552–8560.
   Web of Science ®Google Scholar
• Srivastava, H. M., Özger, F., Mohiuddine, S. A. (2019). Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter λ. Symmetry. 11(3):316. DOI: 10.3390/sym11030316.
   Google Scholar
• Qin, X., Hu, G., Zhang, N., Shen, X., Yang, Y. (2013). A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters. Appl. Math. Comput. 223:1–16. DOI: 10.1016/j.amc.2013.07.073.
   Web of Science ®Google Scholar
• Bohman, H. (1952). On approximation of continuous and analytic functions. Ark. Mat. 2(1):43–56. DOI: 10.1007/BF02591381.
   Google Scholar
• Korovkin, P. P. (1960). On convergence of linear positive operators in the space of continuous. Dokl. Akad. Nauk.
   Web of Science ®Google Scholar
• Altomare, F., Campiti, M. (1994). Korovkin-type approximation theory and its applications. De Gruyter studies in mathematics, Vol. 17, p 627. Berlin, New York: Walter de Gruyter.
   Google Scholar
• Kadak, U. (2018). Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem. J. Math. Anal. Appl. 468 (1):38–63. DOI: 10.1016/j.jmaa.2018.06.047.
   Web of Science ®Google Scholar
• Kadak, U., Dutta, H. (2019). Generalized double statistical weighted summability and its application to Korovkin Type approximation theorem. In: Advanced Topics in Mathematical Analysis. pp. 219–244.
   Google Scholar
• Ditzian, Z., Totik, V. (1987). Moduli of smoothness. New York: Springer-Verlag.
   Google Scholar
• Finta, Z. (2011). Remark on Voronovskaja theorem for q-Bernstein operators. Stud. Univ. Babes-Bolyai Math. 56:335–339.
   Google Scholar
• Acar, T. (2016). Quantitative q-Voronovskaya and q-Grüss-Voronovskaya-type results for q-Szász operators. Georgian Math. J. 23(4):459–468. DOI: 10.1515/gmj-2016-0007.
   Web of Science ®Google Scholar
• Acu, A. M., Gonska, H., Raşa, I. (2011). Grüss-type and Ostrowski-type inequalities in approximation theory. Ukr. Math. J. 63(6):843–864. DOI: 10.1007/s11253-011-0548-2.
   Web of Science ®Google Scholar
• Gal, S. G., Gonska, H. (2015). Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7:97–122.
   Google Scholar
• Baliarsingh, P., Kadak, U., Mursaleen, M. (2018). On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems. Quaest. Math. 41(8):1117–1133. DOI: 10.2989/16073606.2017.1420705.
   Web of Science ®Google Scholar
• Gadjiev, A. D., Orhan, C. (2002). Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32(1):129–138. DOI: 10.1216/rmjm/1030539612.
   Web of Science ®Google Scholar
• Kadak, U. (2019). Generalized statistical convergence based on fractional order difference operator and its applications to approximation theorems. Iran. J. Sci. Technol. Trans. Sci. 43(1):225–237. DOI: 10.1007/s40995-017-0400-0.
   Web of Science ®Google Scholar
• Fast, H. (1951). H. Sur la convergence statistique. Colloq. Math. 2(3–4):241–244. DOI: 10.4064/cm-2-3-4-241-244.
   Google Scholar
• Altay, B., Başar, F., Malkowsky, E. (2009). Matrix transformations on some sequence spaces related to strong Cesaro summability and boundedness. Appl. Math. Comput. 211(2):255–264. DOI: 10.1016/j.amc.2009.01.062.
   Web of Science ®Google Scholar
• Duman, O., Orhan, C. (2008). Rates of A-statistical convergence of operators in the space of locally integrable functions. Appl. Math. Lett. 21(5):431–435. DOI: 10.1016/j.aml.2007.05.017.
   Web of Science ®Google Scholar
• Edely, O. H. H., Mursaleen, M. (2009). On statistical A-summability. Math. Comput. Model. 49(3-4):672–680. DOI: 10.1016/j.mcm.2008.05.053.
   Web of Science ®Google Scholar
• Kadak, U. (2018). Weighted statistical relative invariant mean in modular function spaces with related approximation results. Numer. Funct. Anal. Optim. 39(11):1181–1207. DOI: 10.1080/01630563.2018.1470096.
   Web of Science ®Google Scholar
• De Vore, R. A., Lorentz, G. G. (1993). Constructive approximation. Berlin: Springer.
   Google Scholar
• Dalmanoglu, D. O. (2010). On statistical approximation properties of Kantorovich type q-Bernstein operators. Math. Compu. Model. 52(5–6):760–771.
   Google Scholar
• Radu, C. (2008). Statistical approximation properties of Kantorovich operators based on q-integers. Creat. Math. Inform. 17(2):75–84.
   Google Scholar
• Ren, M. Y., Zeng, X. M. (2014). Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators. J. Ineq. Appl. 2014(10). DOI: 10.1186/1029-242X-2014-10.
   Web of Science ®Google Scholar