  We prove matching direct and inverse theorems for (algebraic) polynomial
approximation with doubling weights $w$ having finitely many zeros and
singularities (i.e., points where $w$ becomes infinite) on an interval and not
too ``rapidly changing'' away from these zeros and singularities. This class of
doubling weights
  is rather wide and, in particular, includes the classical Jacobi weights,
generalized Jacobi weights and generalized Ditzian-Totik weights.
  We approximate in the weighted $L_p$ (quasi) norm $\|f\|_{p, w}$ with
$0<p<\infty$, where $\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) du
\right)^{1/p}$. Equivalence type results involving related realization
functionals are also discussed.
