Polynomial approximation with doubling weights having finitely many zeros and singularities

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.