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 $\mathrm{\parallel }pf-1{\mathrm{\parallel }}_{\alpha }$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\mathrm{\parallel }{p}_{n}f-1{\mathrm{\parallel }}_{\alpha }$ as $n$ approaches $\mathrm{\infty }$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.