1
$\begingroup$

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

$\endgroup$
2

1 Answer 1

2
$\begingroup$

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 $f.$

Or, sledgehammer proof: apply Mergelyan's theorem.

$\endgroup$
1
  • $\begingroup$ Mergelyan? That's a nuke. $\endgroup$
    – zhw.
    Feb 20, 2019 at 22:18

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .