Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

Details Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

2009-03-21

arxiv.org/abs/0903.3652v1

Constr. Approx. (2009) 29; 23-39

Mathematics

Classical Analysis and ODEs

Scientific paper

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$ on $[-1,-a]\cup[a,1]$ by Laurent polynomials.

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