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 ww having finitely many zeros and singularities (i.e., points where ww 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 LpLp (quasi) norm ‖f‖p,w‖f‖p,w with 0