Cyclicity in Dirichlet-type spaces and extremal polynomials

Abstract

For functions f in Dirichlet-type spaces D α, we study how to determine constructively optimal polynomials p n that minimize (Formula presented.) among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of (Formula presented.) 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.