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We prove matching direct and inverse theorems for (algebraic) polynomialapproximation with doubling weights $w$ having finitely many zeros andsingularities (i.e., points where $w$ becomes infinite) on an interval and nottoo ``rapidly changing'' away from these zeros and singularities. This class ofdoubling 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 realizationfunctionals are also discussed.

Polynomial approximation with doubling weights having finitely many zeros and singularities