Progress in Approximation Theory pp 431-451 | Cite as

# Asymptotics of Weighted Polynomials

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## Abstract

We survey recent developments in the theory of the *weighted polynomials w*(*x*)^{ n } *P* _{ n }(*x*), *P* _{ n } ∈ *P* _{ n } on a closed set A ⊂ **R**, with a continuous weight *w*(*x*) ≥ 0 on *A*. Important questions are: Where are the extreme points of the weighted polynomials distributed on *A*, in particular the alternation points of the weighted Chebyshev polynomials *w* ^{ n } *C* _{ w } _{,n }? Which continuous functions *f* on *A* are approximable by the weighted polynomials? How do the polynomials *P* _{ n } of weighted norm *w* ^{ n } *P* _{ n } _{ C } _{(A)} = 1 behave outside of *A*?

This is based on our own work (in particular, in the first two sections) and on work of Mhaskar and Saff, and others.

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