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

Date
2010
Authors
Ostrovskii I.
Ostrovska, S.
Advisor
Instructor
Source Title
Applied Mathematics and Computation
Print ISSN
0096-3003
Electronic ISSN
Publisher
Volume
217
Issue
1
Pages
65 - 72
Language
English
Type
Article
Journal Title
Journal ISSN
Volume Title
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.

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Book Title
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
Citation
Published Version (Please cite this version)