On the analyticity of functions approximated by their q-Bernstein polynomials when q > 1
Author
Ostrovskii I.
Ostrovska, S.
Date
2010Source Title
Applied Mathematics and Computation
Print ISSN
0096-3003
Volume
217
Issue
1
Pages
65 - 72
Language
English
Type
ArticleItem Usage Stats
122
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views
93
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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 continuationAnalytic 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/11693/22242Published Version (Please cite this version)
http://dx.doi.org/10.1016/j.amc.2010.04.020Collections
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