Weighted one-sided approximation of characteristic functions of intervals by polynomials on a closed interval

We consider the problem of weighted one-sided approximation on the interval $[-1,1]$ of characteristic functions of intervals $(a,1]\subset(-1,1]$ and $(a,b)\subset(-1,1)$ by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate difficulties arising in the case of an open interval.