Let be a closed subset on the real line and a nonnegative continuous function on , such that as if is unbounded. Let be the set of functions for which there exists a sequence of weighted polynomials converging to uniformly on . Here is a polynomial of degree at most . is a subalgebra of Let be the closed subset of , such that if and only if is continuous on and vanishes on . The non-trivial approximation of is possible only on the support of an extremal measure that solves an associated equilibrium problem and is smooth, i.e.
In the paper, the following results are shown.
1. If does not belong to then is smooth on some neighborhood of
2. Suppose that is smooth on Then if one of the following conditions holds.
a) can be written as the union of finitely many intervals and the restriction of to each is a doubling measure on .
b) has a positive lower bound in a neighborhood
As corollaries, the authors obtain all previous results for approximation as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.