Complex Operators Generated by q-Bernstein Polynomials, q≥1
Keywords:
q-Bernstein-type operator, Voronovskaja's theorem, quantitative estimates, complex rational operators, complex trigonometric polynomialsAbstract
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.References
Cárdenas-Morales, D., Garrancho, P., Ra¸sa I., Bernstein-type operators which preserve polynomials. Comput. Math. Appl., 62(2011), no. 1, 158163.
Gal, S.G., Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, vol. 8, World Scienti c 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, 163186.
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 disksin the case q > 1, Appl. Math. Comp., 237(2014), 293303.
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, 6572.
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 Mi in 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), 744750.
