Modified ρ-Bernstein Operators for Functions of Two Variables

References

• 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
• Acar, T., Kajla, A. (2018). Degree of approximation for bivariate generalized Bernstein type operators. Results Math. 73(2):79. doi.org/10.1007/s00025-018-0838-1. DOI: 10.1007/s00025-018-0838-1.
   Web of Science ®Google Scholar
• Kajla, A., Acar, T. (2018). Blending type approximation by generalized Bernstein-Durrmeyer type operators. Miskolc Math. Notes. 19(1):319–336. DOI: 10.18514/MMN.2018.2216.
   Web of Science ®Google Scholar
• Mohiuddin, S. A., Acar, T., Alotaibi, A. (2017). Construction of a new family of Bernstein-Kantorovich operators. Math. Methods Appl. Sci. 40(18):7749–7759. DOI: 10.1002/mma.4559.
   Web of Science ®Google Scholar
• Acu, A. M., Agrawal, P. N., Neer, T. (2017). Approximation properties of the modified Stancu operators. Numer. Funct. Anal. Optim. 38(3):279–292. DOI: 10.1080/01630563.2016.1248564.
   Web of Science ®Google Scholar
• Acu, A. M., Hodiş, S., Raşa, I. (2018). A survey on estimates for the differences of positive linear operators. Constr. Math. Anal. 1(2):113–127. DOI: 10.33205/cma.478408.
   Google Scholar
• Acu, A. M., Raşa, I. (2016). New estimates for the differences of positive linear operators. Numer. Algor. . 73(3):775–789. DOI: 10.1007/s11075-016-0117-8.
   Web of Science ®Google Scholar
• Acar, T., Aral, A., Rasa, I. (2016). The new forms of Voronovskaya’s theorem in weighted spaces. Positivity. 20(1):25–40. DOI: 10.1007/s11117-015-0338-4.
   Web of Science ®Google Scholar
• Aral, A., Gupta, V., Agarwal, R. P. (2013). Applications of q-Calculus in operator theory. New York, NY: Springer.
   Google Scholar
• Bǎrbosu, D. (2018). On the remainder term of some bivariate approximation formulas based on linear and positive operators. Constr. Math. Anal. 1(2):73–87.
   Google Scholar
• Bögel, K. (1962). Über die mehrdimensionale differentiation. Jahresber. Deutsch. Math.-Verein. 65:45–71.
   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. DOI: 10.1016/j.amc.2010.01.099.
   Web of Science ®Google Scholar
• Gal, S. G., Trifa, S. (2019). Quantitative estimates for Lp-approximation by Bernstein-Kantorovich-Choquet polynomials with respect to distorted lebesgue measures. Constr. Math. Anal. 2(1):15–21.
   Google Scholar
• Gavrea, I., Ivan, M. (2012). An answer to a conjecture on Bernstein operators. J. Math. Anal. Appl. 390(1):86–92. DOI: 10.1016/j.jmaa.2012.01.021.
   Web of Science ®Google Scholar
• Gonska, H. (2007). On the degree of approximation in Voronovskaja’s theorem. Studia Univ. Babes Bolyai Math. 52:103–115.
   Google Scholar
• Gonska, H. H. (1985). Quantitative Approximation in C(X), Habilitationsschrift. Germany: Universitd˙a Duisburg.
   Google Scholar
• Gupta, V., Agarwal, R. P. (2014). Convergence Estimates in Approximation Theory. Cham: Springer
   Google Scholar
• Mursaleen, M., Ansari, K. J., Khan, A. (2015). On (p, q)-analogue of Bernstein operators. Appl. Math. Comput. 266:874–882. Erratum to On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 278 (2016) 70-71.
   Web of Science ®Google Scholar
• Nowak, G. (2009). Approximation properties for generalized q-Bernstein polynomials. J. Math. Anal. Appl. 350(1):50–55. DOI: 10.1016/j.jmaa.2008.09.003.
   Web of Science ®Google Scholar
• Păltănea, R. (2004). Approximation Theory Using Positive Linear Operators. Boston: Birkhäuser.
   Google Scholar
• Tachev, G. (2012). The complete asymptotic expansion for Bernstein operators. J. Math. Anal. Appl. 385(2):1179–1183. DOI: 10.1016/j.jmaa.2011.07.042.
   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
• Gupta, V., Tachev, G., Acu, A. M. (2019). Modified Kantorovich operators with better approximation properties. Numer. Algor., 81:125–149. DOI: 10.1007/s11075-018-0538-7.
   Web of Science ®Google Scholar
• Acu, A. M., Acar, T., Muraru, C. V., Radu, V. A. (2019). Some approximation properties by a class of bivariate operators. Math. Methods Appl. Sci. 42:5551–5565. DOI: 10.1002/mma.5515.
   Web of Science ®Google Scholar
• Kajla, A., Acar, T. (2019). Modified α-Bernstein operators with better approximation properties. Ann. Funct. Anal. 10(4):570–582. DOI: 10.1215/20088752-2019-0015.
   Web of Science ®Google Scholar
• Bǎrbosu, D. (2000). Some generalized bivariate Bernstein operators. Miskolc Math. Notes. 1:3–10.
   Google Scholar
• Bǎrbosu, D. Aproximarea funcţiilor de mai multe variabile prin sume booleene de operatori liniari de tip interpolator, Cluj- Napoca: Ed Risoprint, (2002). (in Romanian).
   Google Scholar
• Delvos, F. J., Schempp, W. (1989). Boolean methods in interpolation and approximation. Harlow, U.K.: Longman Scientific and Technical.
   Google Scholar
• Pop, O. T., Bărbosu, D. (2008). GBS operators of Durrmeyer- Stancu type. Miskolc Math. Notes. 9(1):53–60. DOI: 10.18514/MMN.2008.133.
   Web of Science ®Google Scholar
• Volkov, V. I. (1957). On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk, SSSR (N.S). 115:17–19.
   Google Scholar
• Pop, O. T., Farkas, M. D. (2009). About the bivariate operators of Kantorovich type. Acta Math. Univ. Comenianae. LXXVIII(1):43–52.
   Google Scholar
• Butzer, P. L., Berens, H. (1967). Semi-Groups of Operators and Approximation, Berlin, Heidelberg: Springer.
   Google Scholar
• Bögel, K. (1934). Mehrdimensionale Differentiation von Funtionen mehrerer Veränderlicher. J. Reine Angew. Math. 170:197–217.
   Google Scholar
• Bögel, K. (1935). Über die mehrdimensionale differentiation, integration und beschränkte variation. J. Reine Angew. Math. 173:5–29.
   Google Scholar
• Dobrescu, E., Matei, I. (1966). The approximation by Bernstein type polynomials of bidimensionally continuous functions, (Romanian). An. Univ. Timişoara Ser. Sti. Mat.-Fiz. 4:85–90.
   Google Scholar
• Badea, C., Badea, I., Cottin, C., Gonska, H. H. (1988). Notes on the degree of approximation of B-continuous and B-differentiable functions. J. Approx. Theory Appl. 4:95–108.
   Google Scholar
• Agrawal, P. N., Ispir, N. (2016). Degree of approximation for bivariate Chlodowsky-Szász-Charlier type operators. Results Math. 69(3–4):369–385. DOI: 10.1007/s00025-015-0495-6.
   Web of Science ®Google Scholar
• Gupta, V., Rassias, T. M., Agrawal, P. N., Acu, A. M. (2018). Recent Advaces in Constructive Approximation Theory. Cham: Springer.
   Google Scholar
• Badea, C., Badea, I., Gonska, H. H. (1986). A test function theorem and approximation by pseudopolynomials. Bull. Austral. Math. Soc. 34(1):53–64. DOI: 10.1017/S0004972700004494.
   Web of Science ®Google Scholar
• Bǎrbosu, D., Acu, A. M., Muraru, C. V. (2017). On certain GBS- Durrmeyer operators based on q- integers. Turk. J. Math. 41:306–380.
   Web of Science ®Google Scholar
• Bărbosu, D., Muraru, C. V. (2015). Approximating B-continuous functions using GBS operators of Bernstein-Schurer-Stancu type based on q-integers. Appl. Math. Comput. 259:80–87. DOI: 10.1016/j.amc.2015.02.030.
   Web of Science ®Google Scholar
• Pop, O. T., Farkas, M. D. (2006). Approximation of B- continuous and B- differentiable functions by GBS operators of Bernstein bivariate polynomials. J. Inequal. Pure Appl. Math. 7(3):9.
   Google Scholar
• Pop, O. T. (2007). Approximation of B-differentiable functions by GBS operators. Anal. Univ. Oradea, Fasc. Matem. Tom XIV:15–31.
   Google Scholar