Cyclicity in Dirichlet-type spaces and extremal polynomials

Abstract: For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $|p f-1|\alpha$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $|p{n}f-1|_{\alpha}$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.