Distribution of alternation points in uniform polynomial approximation

Lorentz, G. G.

doi:10.1090/S0002-9939-1984-0759662-2

Distribution of alternation points in uniform polynomial approximation
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by G. G. Lorentz PDF

Proc. Amer. Math. Soc. 92 (1984), 401-403 Request permission

Abstract:

For a continuous function $f$ on $\left [ {0,1} \right ]$, we discuss the points where the polynomial ${P_n}\left ( x \right )$ of best uniform approximation deviates most from $f\left ( x \right )$, and the signs of the difference $f\left ( x \right ) - {P_n}\left ( x \right )$ alternate. We show that these points can be very irregularly distributed in $\left [ {0,1} \right ]$, even if $f$ is entire.

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