# When does $P_n \to\delta$ implies that $f *P_n \to f$ in $L^\infty(\mathbb T)$?

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?

• 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. – Giuseppe Negro Oct 22 at 14:07
• @GiuseppeNegro Oh, I am asking for general $P_n$ that converges to $\delta$. The Dirichlet kernel is just one example. – eigenvalue Oct 22 at 14:36
• The dense subset of $\mathbb{T}$ on which $P_n * f$ does not converge, is that a null-set? – md2perpe Oct 22 at 22:01