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APPROXIMATION BY BERNSTEIN POLYNOMIALS AT THE POINT OF DISCONTINUITY

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Date Issued:
2011
Abstract/Description:
Chlodovsky showed that if x0 is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bn(f; x0) converge to the average of the one-sided limits on the right and on the left of the function f at the point x0. In 2009, Telyakovskii in extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same result fails for the odd order case. Then in 2010, Tonkov in found the right formulation and proved the result that was missing in the odd-order case. It turned out that the limit in the odd order case is related to the jump of the highest derivative. The proofs in these two cases look similar but have many subtle differences, so it is desirable to find out if there is a unifying principle for treating both cases. In this thesis, we obtain a unified formulation and proof for the asymptotic results of both Telyakovskii and Tonkov and discuss extension of these results in the case where the highest derivative of the function is only assumed to be bounded at the point under study.
Title: APPROXIMATION BY BERNSTEIN POLYNOMIALS AT THE POINT OF DISCONTINUITY.
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Name(s): Liang, Jie, Author
Li, Xin, Committee Chair
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2011
Publisher: University of Central Florida
Language(s): English
Abstract/Description: Chlodovsky showed that if x0 is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bn(f; x0) converge to the average of the one-sided limits on the right and on the left of the function f at the point x0. In 2009, Telyakovskii in extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same result fails for the odd order case. Then in 2010, Tonkov in found the right formulation and proved the result that was missing in the odd-order case. It turned out that the limit in the odd order case is related to the jump of the highest derivative. The proofs in these two cases look similar but have many subtle differences, so it is desirable to find out if there is a unifying principle for treating both cases. In this thesis, we obtain a unified formulation and proof for the asymptotic results of both Telyakovskii and Tonkov and discuss extension of these results in the case where the highest derivative of the function is only assumed to be bounded at the point under study.
Identifier: CFH0004099 (IID), ucf:44790 (fedora)
Note(s): 2011-12-01
B.S.
Sciences, Dept. of Mathematics
Bachelors
This record was generated from author submitted information.
Subject(s): Bernstein polynomials
approximation
asymptotic formulas
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFH0004099
Restrictions on Access: public
Host Institution: UCF

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