The Complex Form of a New Generalization of the Bernstein Operator, Depending On a Non-Negative Real Parameter

References

[1] Altomare F., Korovkin-type Theorems and Approximation by Positive Linear Operators, Surveys in Approximation Theory, 5, 92-164, (2010).

[2] Anastassiou, G.A. and Gal, S.G., Approximation by Complex Bernstein-Schurer and Kantorovich-Schurer Polynomials in Compact Disks, Computer & Mathematics with Applications, 58, 734-743, (2009).

[3] Apostol, T.M., Mathematical Analysis, Second Edition, Wesley Publishing Company, (1974).

[4] Brannan D., A First Course in Mathematical Analysis, Cambridge University Press, (2006).

[5] Bustamante J., Bernstein Operators and Their Properties, Springer, Berlin, (2017).

[6] Bustamante, J., Quesada, J.M., A Property of Ditzian-Totik Second Order Moduli, Applied Mathematics Letter, 23 No. 5, 576-580, (2010).

[7] Burkill J.C., A First Course in Mathematical Analysis, Fifth Edition, Cambridge University Press, (1974).

[8] Cetin N., Approximation and Geometric Properties of Complex - Bernstein Operator, Result in Mathematics, 40, 1-16, (2019).

[9] Cetin N., A new Complex Generalized Bernstein-Schurer operator, Carpathian Journal of Mathematics, 37, 81-89, (2021).

[10] Chen, X., Tan, J., Liu, Z., Xie, J., Approximation of Functions by a New Family of Generalized Bernstein operators. Journal of Mathematical Analysis and Applications, 450, 244-261, (2017).

[11] Devore R. A., The Approximation of Continuous Functions by Positive Linear Operators, Springer, Berlin, (1972).

[12] Devore, R., Lorentz, G.G., Constructive Approximation. Springer, Berlin, (1993).

[13] Friedberg S., Insel A., Spence L., Linear Algebra, Fourth Edition, Pearson Education Limited, (2014).

[14] Gal, S.G., Approximation by Complex Bernstein and Convolution Type Operators, World Scientific Publishing Company Private Limited, Hackensack, New Jersey, (2009).

[15] Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, Pure and Applied Mathematics, Marcel Dekker, New York, (2003).

[16] Gupta R. K., Numerical Methods Fundamentals and Applications, Cambridge University Press, (2019).

[17] Jackson D., The Theory of Approximation, American Mathematical Society Colloquium Publications, (1930).

[18] Jiang, B. and Yu, D., On approximation by Stancu type Bernstein- Schurer polynomials in compact disks, Results in Mathematics, 72, (2017), 1623–1638

[19] John B. C., A First Course in Analysis, Cambridge University Press, (2018).

[20] Kreyszig, E., Introdctory Functional Analysis with Application, John Wiley, (2007).

[21] Lipschutz S., Schaum’s Outlines General Topology, McGraw-Hill Book Company, (1965).

[22] Lorentz, G.G., Bernstein Polynomials, Second Edition, Chelsea Publications, New York, (1986).

[23] Morris P. J., Ordinary Differential Equations, Nabu Press, (2010).

[24] Paltanea R., Approximation Theory Using Positive Linear Operators, Springer, Berlin, (2004).

[25] Ponnusamy S., Silverman H., Complex Variables with Applications, Birkhauser, Berlin, (2006).

[26] Royden H.L., Fitzpatrick P.M., Real Analysis, Pearson Education Incarnation Publishing, Fourth Edition, New York, (1993).

[27] Rudin W., Principles of Mathematical Analysis, Third Edition, McGraw-Hill, (1976).

[28] Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 No.2, 211-219, (1983).

[29] Yang, R., Xiong, J. and Cao, F., Multivariate Stancu operators defined on a simplex, Applied Mathematics & Computation, 138, 189–198, (2003).

@article{1716749,
      author = {Amish Kumar, Shivani Sharma, Dipika Gautam, Chandra Mukesh Swami},
      title = {The Complex Form of a New Generalization of the Bernstein Operator, Depending On a Non-Negative Real Parameter},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {9},
      number = {10},
      pages = {3302-3331},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1716749.pdf},
      abstract = {In this chapter we discuss the complex form of a new generalization of the Bernstein operator, depending on a non-negative real parameter. In which we obtained quantitative upper estimate for simultaneous approximation. In this chapter, we discussed the complex form of a new generalization of the Bernstein operator, depending on a non-negative real parameter and obtained quantitative upper estimate for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation. Also, we present some shape preserving properties of the complex _-Bernstein operator such as univalence, starlikeness, convexity and spirallikeness. We obtained quantitative upper estimate for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation.},
      month = {April},
      doi = {https://doi.org/10.64388/IREV9I10-1716749}
  }