Asymptotics of the Best Polynomial Approximation of vertical bar chi vertical bar (p) and of the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

摘要

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 vertical bar x vertical bar (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]a(a)[a,1] by Laurent polynomials.