Complex Operators Generated by q-Bernstein Polynomials, q≥1
Dedicated to the memory of Akif D. Gadjiev
Keywords:
q-Bernstein-type operator, Voronovskaja's theorem, quantitative estimates, complex rational operators, complex trigonometric polynomials.Abstract
By using an univalent and analytic function τ in a suitable open disk centered in origin, we attach to analytic functions f, the complex Bernstein-type operators of the form B_{n,q}^{τ}(f)=B_{n,q}(f∘τ⁻¹)∘τ , where B_{n,q} denote the classical complex q-Bernstein polynomials, q≥1. The new complex operators satisfy the same quantitative estimates as B_{n,q}. As applications, for two concrete choices of τ, we construct complex rational functions and complex trigonometric polynomials which approximate f with a geometric rate.
Mathematics Subject Classification (2010): 30E10, 41A35, 41A25.
References
Cardenas-Morales, D., Garrancho, P., Rasa I., Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62(2011), no. 1, 158-163.
Gal, S.G., Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, vol. 8, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009.
Goodman, A.W., An invitation to the study of univalent and multivalent functions, Internat. J. Math. Math. Sci., 2(1979), no. 2, 163-186.
Mahmudov, N., Kara, M., Approximation theorems for generalized complex Kantorovich-type operators, J. Appl. Math., (2012), Article Number: 454579.
Mahmudov, N., Approximation by q-Durrmeyer type polynomials in compact disks in the case q > 1, Appl. Math. Comp., 237(2014), 293-303.
Ostrovskii, I., Ostrovska, S., On the analyticity of functions approximated by their q-Bernstein polynomials when q > 1, Appl. Math. Comp., 217(2010), no. 1, 65-72.
Ostrovska, S., q-Bernstein polynomials and their iterates, J. Approx. Theory, 123(2003), 232-255.
Sveshnikov, A.G., Tikhonov, A.N., The Theory of Functions of a Complex Variable, Mir Publishers, Moscow, 1971.
Silverman, H., Complex Variables, Houghton Miflin Co., Boston, 1975.
Wang, H., Wu, X.Z., Saturation of convergence for q-Bernstein polynomials in the case q _ 1, J. Math. Anal. Appl., 337(2008), 744-750.
Downloads
Published
Issue
Section
License
Copyright (c) 2016 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
