# Let f be analytic in the open unit disk and continuous on the boundary is there a sequence of polynomials converging uniformly to f?

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?

The functions $$f_t$$ defined by $$f_t(z)=f(tz)$$ for $$t<1$$ converge uniformly on the unit disc to $$f$$ as $$t\to 1.$$ And each $$f_t$$ is analytic on $$\{z\mid |z|<1/t\}$$ so its Taylor series converges uniformly to $$f_t$$ on the unit disc. So by an $$\epsilon/2$$ argument there is a series of polynomials converging uniformly to $$f.$$