Existence of a power series converging non-uniformly to a continuous function

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.

Solutions Collecting From Web of "Existence of a power series converging non-uniformly to a continuous function"

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.