Let $f$ be analytic in the open unit disk and continuous on the boundary is there a sequence of polynomials converging uniformly to $f$? Obviously since $f$ is analytic on the open disk we know for sure that in any disk or radius smaller than $1$ the taylor series of $f$ around $0$ converges uniformly however we cannot tell if the convergence of the taylor series is gonna be uniform on the circle of radius $1$. It will certainly converge to $f$ because $f$ is continuous but we cannot tell if its gonna be uniform. So is there anyway to construct a sequence of polynomials that converge uniformly to $f$ in the closed unit disk?
Thanks in advance