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

Question

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?