Cyclicity in Dirichlet-type spaces and extremal polynomials

Abstract

For functions f in Dirichlet-type spaces $${D_\alpha }$$ , we study how to determine constructively optimal polynomials p n that minimize $${\left\| {pf - 1} \right\|_\alpha }$$ among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of $${\left\| {{p_n}f - 1} \right\|_\alpha }$$ 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.