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      On the analyticity of functions approximated by their q-Bernstein polynomials when q > 1

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      Author
      Ostrovskii I.
      Ostrovska, S.
      Date
      2010
      Source Title
      Applied Mathematics and Computation
      Print ISSN
      0096-3003
      Volume
      217
      Issue
      1
      Pages
      65 - 72
      Language
      English
      Type
      Article
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      Abstract
      Since in the case q > 1 the q-Bernstein polynomials Bn,q are not positive linear operators on C[0, 1], the investigation of their convergence properties for q > 1 turns out to be much harder than the one for 0 < q < 1. What is more, the fast increase of the norms ∥Bn,q∥ as n → ∞, along with the sign oscillations of the q-Bernstein basic polynomials when q > 1, create a serious obstacle for the numerical experiments with the q-Bernstein polynomials. Despite the intensive research conducted in the area lately, the class of functions which are uniformly approximated by their q-Bernstein polynomials on [0, 1] is yet to be described. In this paper, we prove that if f:[0,1]→C is analytic at 0 and can be uniformly approximated by its q-Bernstein polynomials (q > 1) on [0, 1], then f admits an analytic continuation from [0, 1] into {z: z < 1}. © 2010 Elsevier Inc. All rights reserved.
      Keywords
      Analytic continuation
      Analytic function
      Q-Bernstein polynomials
      Q-Integers
      Uniform convergence
      Analytic continuation
      Analytic functions
      Analyticity
      Bernstein polynomial
      Convergence properties
      Intensive research
      Numerical experiments
      Positive linear operators
      Uniform convergence
      Amber
      Functional analysis
      Functions
      Mathematical operators
      Polynomials
      Permalink
      http://hdl.handle.net/11693/22242
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.amc.2010.04.020
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      • Department of Mathematics 578

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