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I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:

  • $f$ converges and is continuous on the closed unit disk $D$ and
  • the series $\sum_n a_n z^n$ does not converge uniformly on $D$.

I have tried to construct a counter-example, but with no success so far.

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    $\begingroup$ Related: math.stackexchange.com/q/353760/462 (In the answer to the linked question, an $f$ is given that converges pointwise on the boundary of $D$, but its restriction to the boundary is discontinuous (so the convergence cannot be uniform). $\endgroup$ – Andrés E. Caicedo May 13 '13 at 14:06
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Yes. See D.R. Lick's article, "Sets of non-uniform convergence of Taylor Series."

It is shown there that for every closed subset $F$ of the boundary $\partial D$, there is such a series that converges everywhere on the closed disk to a continuous function, and whose set of non-uniform convergence is $F$. This means that the series converges uniformly in a neighborhood (open arc) of each point in $\partial D\setminus F$, but not in any neighborhood of any point in $F$.

I have been interested in this question before, which is why I have a reference handy. I referred to this article in another answer, to a question about power series in Banach algebras. As mentioned there, the sequence of Cesàro means of the power series will converge uniformly on the closed disk.

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