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.
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