Abstract
For functions f in Dirichlet-type spaces D-alpha, we study how to determine constructively optimal polynomials p(n) that minimize parallel to pf - 1 parallel to(alpha) among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of parallel to p(n)f - 1 parallel to(alpha) as n approaches infinity, 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.