In the question $\mathbb T$ is a unit circle. For one example, even the continuity of $f$ does not suffice. If we let $P_n$ be the Dirichlet kernel $$P_n=\frac{1}{2\pi} \sum_{m=-n}^n e^{imx},$$ which goes to $\delta$ as $n\to\infty$, then one can prove that there is a continuous function whose Fourier series $P_n * f$ does not converge uniformly(even pointwise in a dense subset of $\mathbb T$) to $f$. Then what additional condition is needed to the question?

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    $\begingroup$ This is just a complicated way of asking "when does a Fourier series converge uniformly?" This is a classical question and you can find some insight on Wikipedia, for a start. $\endgroup$ – Giuseppe Negro Oct 22 at 14:07
  • $\begingroup$ @GiuseppeNegro Oh, I am asking for general $P_n$ that converges to $\delta$. The Dirichlet kernel is just one example. $\endgroup$ – eigenvalue Oct 22 at 14:36
  • $\begingroup$ The dense subset of $\mathbb{T}$ on which $P_n * f$ does not converge, is that a null-set? $\endgroup$ – md2perpe Oct 22 at 22:01

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