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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The degree of coconvex polynomial approximation


Authors: K. Kopotun, D. Leviatan and I. A. Shevchuk
Journal: Proc. Amer. Math. Soc. 127 (1999), 409-415
MSC (1991): Primary 41A10, 41A17, 41A25, 41A29
MathSciNet review: 1459130
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Abstract: Let $f\in C[-1,1]$ change its convexity finitely many times in the interval, say $s$ times, at $Y_{s}:-1<y_{1}<\cdots <y_{s}<1$. We estimate the degree of approximation of $f$ by polynomials of degree $n$, which change convexity exactly at the points $Y_{s}$. We show that provided $n$ is sufficiently large, depending on the location of the points $Y_{s}$, the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of $f$ multiplied by a constant $C(s)$, which depends only on $s$.


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Additional Information

K. Kopotun
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: kkopotun@math.vanderbilt.edu

D. Leviatan
Affiliation: School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: leviatan@math.tau.ac.il

I. A. Shevchuk
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 252601, Ukraine
Email: shevchuk@dad.imath.kiev.ua

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04452-4
PII: S 0002-9939(99)04452-4
Keywords: Coconvex polynomial approximation, Jackson estimates
Received by editor(s): May 9, 1996
Received by editor(s) in revised form: April 1, 1997
Additional Notes: The first author acknowledges partial support by the Izaak Walton Killam Memorial Scholarship.
The second author acknowledges partial support by ONR grant N00014-91-1076 and by DoD grant N00014-94-1-1163.
The third author was partially supported by the State Fund for Fundamental Research of Ukraine.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society