Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions

Let G be a bounded region with simply connected closure G¯ and analytic boundary and let μ be a positive measure carried by G¯ together with finitely many pure points outside G. We provide estimates on the norms of the monic polynomials of minimal norm in the space Lq(μ) for q>0. In case the norms converge to 0, we provide estimates on the rate of convergence, generalizing several previous results. Our most powerful result concerns measures μ that are perturbations of measures that are absolutely continuous with respect to the push-forward of a product measure near the boundary of the unit disk. Our results and methods also yield information about the strong asymptotics of the extremal polynomials and some information concerning Christoffel functions.