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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Piecewise monotone polynomial approximation


Authors: D. J. Newman, Eli Passow and Louis Raymon
Journal: Trans. Amer. Math. Soc. 172 (1972), 465-472
MSC: Primary 41A25; Secondary 41A10
MathSciNet review: 0310506
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Abstract: Given a real function $ f$ satisfying a Lipschitz condition of order 1 on $ [a,b]$, there exists a sequence of approximating polynomials $ \{ {P_n}\} $ such that the sequence $ {E_n} = \vert\vert{P_n} - f\vert\vert$ (sup norm) has order of magnitude $ 1/n$ (D. Jackson). We investigate the possibility of selecting polynomials $ {P_n}$ having the same local monotonicity as $ f$ without affecting the order of magnitude of the error. In particular, we establish that if $ f$ has a finite number of maxima and minima on $ [a,b]$ and $ S$ is a closed subset of $ [a,b]$ not containing any of the extreme points of $ f$, then there is a sequence of polynomials $ {P_n}$ such that $ {E_n}$ has order of magnitude $ 1/n$ and such that for $ n$ sufficiently large $ {P_n}$ and $ f$ have the same monotonicity at each point of $ S$. The methods are classical.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0310506-9
PII: S 0002-9947(1972)0310506-9
Keywords: Monotone approximation, piecewise monotone approximation, Jackson kernel, Jackson's Theorem
Article copyright: © Copyright 1972 American Mathematical Society