Piecewise monotone polynomial approximation

Newman, D. J.; Passow, Eli; Raymon, Louis

doi:10.1090/S0002-9947-1972-0310506-9

Piecewise monotone polynomial approximation
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by D. J. Newman, Eli Passow and Louis Raymon PDF

Trans. Amer. Math. Soc. 172 (1972), 465-472 Request permission

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 ${E_n} = ||{P_n} - f||$ (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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