Consider the Poisson kernel given by $$P_r(\theta)=\sum_{n=-\infty}^\infty r^{|n|}e^{in\theta}=\frac{1-r^2}{(1-r)^2+2r(1-\cos\theta)}$$ Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$, meaning that $f$ is in $L^1$ and is periodic with period $2\pi$. Show that $$\lim_{r\rightarrow 1^-}\int_{[-\pi,\pi]}|f\ast P_r(\theta)-f(\theta)|d\theta=0$$
This kind of statement is similar to the approximate identity lemma, where we have a continuous function with period $2\pi$ and conclude that $|f\ast P_r(\theta)-f(\theta)|$ converges uniformly to $0$ as $r\rightarrow 1^-$.
I'm not sure how to adapt that to this weaker case of $L^1$ function, where the conclusion is also weaker (i.e. convergence in the $L^1$ norm).
