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 of (-1, 1] and (a, b) subset of (-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.