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Let $f:D(0,1) \to \mathbb C$ be continuous on the closed unit disc $D(0,1)$ and holomorphic on the open unit disc.

Show that there exist a sequence of polynomial that converge uniformly in the closed unit disc to $f$.

Any help please?

KTL
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  • This seems false as stated. Try $f(z)=1/(1-z)$. – Ted Shifrin Nov 02 '14 at 00:41
  • @Ted, I think perhaps $D(0,1)$ is intended to be the closed unit disk. So your $f(z)$ is not continuous on $D(0,1)$. – Jair Taylor Nov 02 '14 at 00:50
  • @ Jair Taylor,effectively,$D(0,1)$ is the closed unit disc. – KTL Nov 02 '14 at 00:56
  • Mais remarquez bien que vous avez écrit "the open unit disk." So you mean continuous on the closed disc and holomorphic on the open? Your title and question are somewhat inconsistent and confusing. ;) – Ted Shifrin Nov 02 '14 at 01:12
  • @Ted, thanks for the remark.I just corriged the question. – KTL Nov 02 '14 at 01:27
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    Hint: for each $n$, the function $f_n(z) = f((1-1/n)z)$ is holomorphic in a larger disk, and therefore can be uniformly approximated by polynomials on $D(0,1)$. –  Nov 02 '14 at 07:50
  • Thanks you @JustABrickInTheWall – KTL Nov 02 '14 at 11:11

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