Cyclicity in Dirichlet-type spaces and extremal polynomials

For functions f in Dirichlet-type spaces , we study how to determine constructively optimal polynomials p n that minimize among all polynomials p of degree at most n . We then obtain sharp estimates for the rate of decay of as n approaches ∞, for certain classes of functions f . Finally, inspired by the Brown-Shields conjecture, we prove that certain logarithmic conditions on f imply cyclicity, and we study some computational phenomena pertaining to the zeros of optimal polynomials.