  We discuss approximation of extremal functions by polynomials in the weighted
Bergman spaces $A^p_\alpha$ where $-1 < \alpha < 0$ and $-1 < \alpha < p-2$. We
obtain bounds on how close the approximation is to the true extremal function
in the $A^p_\alpha$ and uniform norms. We also discuss several results on the
relation between the Bergman modulus of continuity of a function and how
quickly its best polynomial approximants converge to it.
